Fermat's Last Theorem (Final Copy)
YogeshKarunavannan
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School of Computing, Engineering and Mathematics
AN INTRODUCTION TO FERMAT’S LAST THEOREM
Yogesh Warren Karunavannan
May 2016
AN INTRODUCTION TO FERMAT’S LAST THEOREM
Yogesh Warren Karunavannan
May 2016
DECLARATION
I declare that no part of the work in this report has been submitted in support of an application
for another degree or qualification at this or any other institute of learning.
Yogesh Warren Karunavannan
I declare that no part of the work in this report has been submitted in support of an application
for another degree or qualification at this or any other institute of learning.
Yogesh Warren Karunavannan
ACKNOWLEDGEMENTS
I am grateful to have had John Taylor supervise the typing of this dissertation and pushing me in
the right direction towards its completion. I would like thank Wajid Mannan for providing me with
essential resources that I needed to type this dissertation. I am also indebted to my friends for
using their spare time to read my work and provide me with valuable suggestions that aided in
the completion of this project.
I am grateful to have had John Taylor supervise the typing of this dissertation and pushing me in
the right direction towards its completion. I would like thank Wajid Mannan for providing me with
essential resources that I needed to type this dissertation. I am also indebted to my friends for
using their spare time to read my work and provide me with valuable suggestions that aided in
the completion of this project.
ABSTRACT
Declared as the ‘most difficult mathematical problem’ in the Guinness Book of World Records,
and only being successfully proven by Andrew Wiles in 1994, Fermat’s Last Theorem was
conjected by Pierre de Fermat in 1637 and has created a great deal of advancements in number
theory. This paper has explored the major steps taken to prove the theorem and focus on the
mathematical techniques used.
Supervisor: John Taylor
Declared as the ‘most difficult mathematical problem’ in the Guinness Book of World Records,
and only being successfully proven by Andrew Wiles in 1994, Fermat’s Last Theorem was
conjected by Pierre de Fermat in 1637 and has created a great deal of advancements in number
theory. This paper has explored the major steps taken to prove the theorem and focus on the
mathematical techniques used.
Supervisor: John Taylor
CONTENTS
1 A brief introduction to Fermat’s Last Theorem ........................................................................... 1
1.1 Creating a general proof .................................................................................................... 1
1.2 The Pythagorean Theorem ................................................................................................. 2
2 The Biquadratic Equation ............................................................................................................ 3
2.1 The Method of Infinite Descent ......................................................................................... 3
2.2 The case of ??????=� ............................................................................................................. 4
2.3 Euclid’s Lemma and the proof of ??????=� ........................................................................... 5
3 The case of ??????=� ....................................................................................................................... 7
Case 1: ?????? is even ........................................................................................................................... 8
Case 2: ?????? is even .......................................................................................................................... 9
Case 1: 3 does not divide into ?????? ................................................................................................ 11
Case 2: 3 divides into ?????? .............................................................................................................. 13
4 Ernst Kummer and the proof for regular primes ....................................................................... 15
4.1 The Cyclotomic ring/field and Root of unity .................................................................... 15
4.2 Gabriel Lamé’s Proposed Proof ........................................................................................ 16
4.3 Ideal Numbers .................................................................................................................. 17
4.4 The proof for Regular Primes ........................................................................................... 19
Case 1: not divisible by ?????? .................................................................................................... 19
Case 2: divisible by ?????? ........................................................................................................... 20
5 Conclusion.................................................................................................................................. 21
6 Evaluation .................................................................................................................................. 22
7 Bibliography ............................................................................................................................... 24
1 A brief introduction to Fermat’s Last Theorem ........................................................................... 1
1.1 Creating a general proof .................................................................................................... 1
1.2 The Pythagorean Theorem ................................................................................................. 2
2 The Biquadratic Equation ............................................................................................................ 3
2.1 The Method of Infinite Descent ......................................................................................... 3
2.2 The case of ??????=� ............................................................................................................. 4
2.3 Euclid’s Lemma and the proof of ??????=� ........................................................................... 5
3 The case of ??????=� ....................................................................................................................... 7
Case 1: ?????? is even ........................................................................................................................... 8
Case 2: ?????? is even .......................................................................................................................... 9
Case 1: 3 does not divide into ?????? ................................................................................................ 11
Case 2: 3 divides into ?????? .............................................................................................................. 13
4 Ernst Kummer and the proof for regular primes ....................................................................... 15
4.1 The Cyclotomic ring/field and Root of unity .................................................................... 15
4.2 Gabriel Lamé’s Proposed Proof ........................................................................................ 16
4.3 Ideal Numbers .................................................................................................................. 17
4.4 The proof for Regular Primes ........................................................................................... 19
Case 1: not divisible by ?????? .................................................................................................... 19
Case 2: divisible by ?????? ........................................................................................................... 20
5 Conclusion.................................................................................................................................. 21
6 Evaluation .................................................................................................................................. 22
7 Bibliography ............................................................................................................................... 24
1
1 A brief introduction to Fermat’s Last Theorem
For almost four centuries, many mathematicians tried to prove Fermat’s famous conjecture, or
otherwise known as his Last Theorem. Pierre de Fermat studied the book Arithmetic by Diophantus in
the 1630’s. During his time, the French lawyer made several notes on the margin of the book. Of all the
notes he made, one of these stood out more than the rest, and translated to English it states, as shown
in Paulo Ribenboim’s book; Fermat's Last Theorem for Amateurs (2013):
“It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general
any power higher than the second into powers of like degree; I have discovered a truly marvellous proof,
which this margin is too small to contain”.
Fermat had a habit of claiming theorems without the proofs, and though all of his conjectures were
proven, the Last Theorem remained unsolved until 1994 by Andrew Wiles. Fermat’s statement and the
theorem, written in a much more visible and familiar way, is as such:
Theorem. If �>2 is any integer. The equation �
�
+�
�
=�
�
has no solution in positive whole
numbers.
Though Fermat claimed to have a ‘marvellous proof’, it was a theorem that many mathematicians failed
to prove for hundreds of years (Singh, 2002, Page 6). Even though the proof exists due to Andrew Wiles,
it inherited many mathematical concepts and techniques that was unavailable in Fermat’s time. We will
explore the introduction and development for the proof of the Last Theorem, such as the proof for
regular primes by Ernst Kummer and the failed proof by Gabriel Lamé, but not include Andrew Wiles’s
proof.
The proofs for this chapter will be largely based on Paulo Ribenboim’s book; Fermat’s Last Theorem for
Amateurs (2013).
1.1 Creating a general proof
To understand Fermat’s Last Theorem in detail, we must focus on the necessary components of the
theorem. In order to obtain a general proof, we must begin with the following:
Theorem 2. To prove Fermat’s Last Theorem for all integers greater than 2, proving that there cannot be
any integer solutions for �=4 and all odd prime numbers, p, is sufficient.
Proof. Let �
�
+�
�
=�
�
have solutions for positive integers �,�,�. Since � is composite, meaning that
it can be divided by numbers other than one and itself, we can confirm that �>2.
1 A brief introduction to Fermat’s Last Theorem
For almost four centuries, many mathematicians tried to prove Fermat’s famous conjecture, or
otherwise known as his Last Theorem. Pierre de Fermat studied the book Arithmetic by Diophantus in
the 1630’s. During his time, the French lawyer made several notes on the margin of the book. Of all the
notes he made, one of these stood out more than the rest, and translated to English it states, as shown
in Paulo Ribenboim’s book; Fermat's Last Theorem for Amateurs (2013):
“It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general
any power higher than the second into powers of like degree; I have discovered a truly marvellous proof,
which this margin is too small to contain”.
Fermat had a habit of claiming theorems without the proofs, and though all of his conjectures were
proven, the Last Theorem remained unsolved until 1994 by Andrew Wiles. Fermat’s statement and the
theorem, written in a much more visible and familiar way, is as such:
Theorem. If �>2 is any integer. The equation �
�
+�
�
=�
�
has no solution in positive whole
numbers.
Though Fermat claimed to have a ‘marvellous proof’, it was a theorem that many mathematicians failed
to prove for hundreds of years (Singh, 2002, Page 6). Even though the proof exists due to Andrew Wiles,
it inherited many mathematical concepts and techniques that was unavailable in Fermat’s time. We will
explore the introduction and development for the proof of the Last Theorem, such as the proof for
regular primes by Ernst Kummer and the failed proof by Gabriel Lamé, but not include Andrew Wiles’s
proof.
The proofs for this chapter will be largely based on Paulo Ribenboim’s book; Fermat’s Last Theorem for
Amateurs (2013).
1.1 Creating a general proof
To understand Fermat’s Last Theorem in detail, we must focus on the necessary components of the
theorem. In order to obtain a general proof, we must begin with the following:
Theorem 2. To prove Fermat’s Last Theorem for all integers greater than 2, proving that there cannot be
any integer solutions for �=4 and all odd prime numbers, p, is sufficient.
Proof. Let �
�
+�
�
=�
�
have solutions for positive integers �,�,�. Since � is composite, meaning that
it can be divided by numbers other than one and itself, we can confirm that �>2.
2
Subsequently we can assume it has a factor � such that � is 4 or an odd prime number, �. Hence there
can exist an exponent � such that �=��, whereby � is an exponent smaller than �. The conclusive
equation is therefore:
�
�
+�
�
=�
�
,
(�
�
)
�
+(�
�
)
�
=(�
�
)
�
.
Thus �
�
,�
�
and �
�
are solutions to the equation with power �, which equals 4 or an odd prime number,
� (Ribenboim, 2013, Page 1-2).
1.2 The Pythagorean Theorem
Fermat’s Last Theorem states that there are no integer solution to the equation �
�
+�
�
=�
�
when
�>2. Yet we know that there are integer solutions to the equation �
2
+�
2
=�
2
or otherwise known
as the Pythagoras equation/theorem. The three positive integers �,�,� that satisfy the equation are
known as the Pythagorean Triples. For example, a triple (�,�,�) such as (3,4,5) can exist since 3
2
+
4
2
=5
2
. In order to create a proof for �=4, we must factorize the equation so that the integers of the
factor are in the form of a Pythagorean Triple whereby ���(�,�,�)=1; this form of factorization is
known as the unique factorization. The abbreviation ��� is used to define the greatest common divisor
(Ribenboim, 2013, Page 3-4).
Theorem 3. If �,� are integers such that �>�>0, ���(�,�)=1. Then the triple (�,�,�), given by:
{
�=2��
�=�
2
−�
2
�=�
2
+�
2
,
is a primitive solution to the equation �
2
+�
2
=�
2
and ���(�,�,�)=1.
Proof. If �,� are integers that satisfy the above equation, then:
�
2
+�
2
=4�
2
�
2
+(�
2
−�
2
)
2
=(�
2
+�
2
)
2
=�
2
.
Since � and � are different in parity, meaning either � or � is positive or negative and �>�>0, then in
relation to the above equation �>0, �>0 and �>0 where � is even and both �, � are odd. Now let
�=���(�,�,�), if � divides into �, � and � then � must divide into 2�
2
and 2�
2
since �+�=2�
2
and
�−�=2�
2
. Therefore �=1 since ���(�,�)=1 and � divides into � and �.
Subsequently we can assume it has a factor � such that � is 4 or an odd prime number, �. Hence there
can exist an exponent � such that �=��, whereby � is an exponent smaller than �. The conclusive
equation is therefore:
�
�
+�
�
=�
�
,
(�
�
)
�
+(�
�
)
�
=(�
�
)
�
.
Thus �
�
,�
�
and �
�
are solutions to the equation with power �, which equals 4 or an odd prime number,
� (Ribenboim, 2013, Page 1-2).
1.2 The Pythagorean Theorem
Fermat’s Last Theorem states that there are no integer solution to the equation �
�
+�
�
=�
�
when
�>2. Yet we know that there are integer solutions to the equation �
2
+�
2
=�
2
or otherwise known
as the Pythagoras equation/theorem. The three positive integers �,�,� that satisfy the equation are
known as the Pythagorean Triples. For example, a triple (�,�,�) such as (3,4,5) can exist since 3
2
+
4
2
=5
2
. In order to create a proof for �=4, we must factorize the equation so that the integers of the
factor are in the form of a Pythagorean Triple whereby ���(�,�,�)=1; this form of factorization is
known as the unique factorization. The abbreviation ��� is used to define the greatest common divisor
(Ribenboim, 2013, Page 3-4).
Theorem 3. If �,� are integers such that �>�>0, ���(�,�)=1. Then the triple (�,�,�), given by:
{
�=2��
�=�
2
−�
2
�=�
2
+�
2
,
is a primitive solution to the equation �
2
+�
2
=�
2
and ���(�,�,�)=1.
Proof. If �,� are integers that satisfy the above equation, then:
�
2
+�
2
=4�
2
�
2
+(�
2
−�
2
)
2
=(�
2
+�
2
)
2
=�
2
.
Since � and � are different in parity, meaning either � or � is positive or negative and �>�>0, then in
relation to the above equation �>0, �>0 and �>0 where � is even and both �, � are odd. Now let
�=���(�,�,�), if � divides into �, � and � then � must divide into 2�
2
and 2�
2
since �+�=2�
2
and
�−�=2�
2
. Therefore �=1 since ���(�,�)=1 and � divides into � and �.
3
2 The Biquadratic Equation
Fermat created his Last Theorem whilst attempting to solve the equation �
4
−�
4
=�
2
. He had the
desire to see if the area of the Pythagorean triangle, would be equal to the square of an integer.
In the margin of Arithmetic, Fermat wrote: "If the area of a right-angled triangle were a square, there
would exist two biquadrates that difference of which would be a square number” (Edwards, 1977, Page
11)
A biquadrate is a value to the fourth-power, therefore, the biquadrate of 2 is 2
4
which is 16. This, in
turn, leads to the equation stated above. By showing that there are no right-angle triangles that have an
area equal to a square, it will, in turn, prove Fermat's Last Theorem for �=4, which will be further
explained in the latter parts of this project.
2.1 The Method of Infinite Descent
Fermat’s infamous method of infinite descent, one which he was greatly proud of suggests that if there
are one or more solutions to an equation, then there can be smaller solutions to that same problem and
so forth. Therefore, if we prove that the smallest solution exists, then it implies that there must be an
even smaller solution. Yet this is impossible, as a sequence of whole positive numbers cannot descent
indefinitely (Edwards, 1977, Page 8). The proof for the method of infinite descent will be based on the
same the proof shown by Carl Eberhart in his paper; Fermat and His Method of Infinite Descent (1999).
For example, let's prove that √2 is irrational. In order to apply the method of infinite descent, we must
first assume that √2 has a rational solution. By applying this assumption, we can show that √2 as a
quotient of two positive integers:
√2=
�1
�1
, where�1>�1.
Now let’s assume the equation:
1
√2−1
=√2+1.
By replacing the √2 on the left-hand side with
�1
�1
, we acquire:
1
√
�1
�1
−1
=√2+1,
2 The Biquadratic Equation
Fermat created his Last Theorem whilst attempting to solve the equation �
4
−�
4
=�
2
. He had the
desire to see if the area of the Pythagorean triangle, would be equal to the square of an integer.
In the margin of Arithmetic, Fermat wrote: "If the area of a right-angled triangle were a square, there
would exist two biquadrates that difference of which would be a square number” (Edwards, 1977, Page
11)
A biquadrate is a value to the fourth-power, therefore, the biquadrate of 2 is 2
4
which is 16. This, in
turn, leads to the equation stated above. By showing that there are no right-angle triangles that have an
area equal to a square, it will, in turn, prove Fermat's Last Theorem for �=4, which will be further
explained in the latter parts of this project.
2.1 The Method of Infinite Descent
Fermat’s infamous method of infinite descent, one which he was greatly proud of suggests that if there
are one or more solutions to an equation, then there can be smaller solutions to that same problem and
so forth. Therefore, if we prove that the smallest solution exists, then it implies that there must be an
even smaller solution. Yet this is impossible, as a sequence of whole positive numbers cannot descent
indefinitely (Edwards, 1977, Page 8). The proof for the method of infinite descent will be based on the
same the proof shown by Carl Eberhart in his paper; Fermat and His Method of Infinite Descent (1999).
For example, let's prove that √2 is irrational. In order to apply the method of infinite descent, we must
first assume that √2 has a rational solution. By applying this assumption, we can show that √2 as a
quotient of two positive integers:
√2=
�1
�1
, where�1>�1.
Now let’s assume the equation:
1
√2−1
=√2+1.
By replacing the √2 on the left-hand side with
�1
�1
, we acquire:
1
√
�1
�1
−1
=√2+1,
4
therefore, by solving for √2, we derive the following equation:
√2=
2−
�1
�1
�1
�1
−1
.
After further simplifying the above equation we obtain:
√2=
2�1−�1
�1−�1
.
Since we know that 1<
�1
�1
<2, due to the above equation, we can denote that both the numerator and
denominator are positive integers. Consequently, we can replace the numerator with the reprehensive
positive integer, �2 and the denominator with �2.
Hence, we have the new quotient of √2:
√2=
�2
�2
, where �2>�2.
We can derive the equation above, since clearly, we know that �2<�1 and �2<�1 due to �1>�1.
We can further repeat this process such that with an infinite amount of repeats, there will exist an
�(�+1), which is less than ��, and a �(�+1), which is less than ��. Henceforth, there is an infinite
descent for all positive integers that satisfy √2=
�
�
, but this is impossible due to the fact that a
sequence of infinitely descending positive integers cannot exist. This, by contradiction, proves that √2 is
irrational and subsequently does not have a quotient of positive integers. As Fermat stressed upon, ‘the
method of infinite descent proves that certain things are impossible’ (Edwards, 1977, Page 8), therefore,
it is also known as the proof by contradiction.
2.2 The case of ??????=�
Now we will take up the case of �=4. In order to show a proof for �=4, we must apply the method of
infinite descent and the method of generating a Pythagorean triple as shown in Theorem 3. The proofs
for �=4 will be largely based on the proofs shown by Larry Freeman in his blog; Fermat’s Last Theorem
(2005) and Harold M. Edwards in his book; Fermat’s Last Theorem: A Genetic Introduction to Algebraic
Number Theory (1977).
Theorem 4. The equation, �
4
+�
4
=�
4
has no positive integer solution �,�,�.
Proof. Using the method to generate a Pythagorean triple, as explained in 1.2, we can assume that
�,�,� have no common divisor greater than 1 and as such, we say that �,�,� are co-prime. The
therefore, by solving for √2, we derive the following equation:
√2=
2−
�1
�1
�1
�1
−1
.
After further simplifying the above equation we obtain:
√2=
2�1−�1
�1−�1
.
Since we know that 1<
�1
�1
<2, due to the above equation, we can denote that both the numerator and
denominator are positive integers. Consequently, we can replace the numerator with the reprehensive
positive integer, �2 and the denominator with �2.
Hence, we have the new quotient of √2:
√2=
�2
�2
, where �2>�2.
We can derive the equation above, since clearly, we know that �2<�1 and �2<�1 due to �1>�1.
We can further repeat this process such that with an infinite amount of repeats, there will exist an
�(�+1), which is less than ��, and a �(�+1), which is less than ��. Henceforth, there is an infinite
descent for all positive integers that satisfy √2=
�
�
, but this is impossible due to the fact that a
sequence of infinitely descending positive integers cannot exist. This, by contradiction, proves that √2 is
irrational and subsequently does not have a quotient of positive integers. As Fermat stressed upon, ‘the
method of infinite descent proves that certain things are impossible’ (Edwards, 1977, Page 8), therefore,
it is also known as the proof by contradiction.
2.2 The case of ??????=�
Now we will take up the case of �=4. In order to show a proof for �=4, we must apply the method of
infinite descent and the method of generating a Pythagorean triple as shown in Theorem 3. The proofs
for �=4 will be largely based on the proofs shown by Larry Freeman in his blog; Fermat’s Last Theorem
(2005) and Harold M. Edwards in his book; Fermat’s Last Theorem: A Genetic Introduction to Algebraic
Number Theory (1977).
Theorem 4. The equation, �
4
+�
4
=�
4
has no positive integer solution �,�,�.
Proof. Using the method to generate a Pythagorean triple, as explained in 1.2, we can assume that
�,�,� have no common divisor greater than 1 and as such, we say that �,�,� are co-prime. The
5
equation above can be written as (�
2
)
2
+(�
2
)
2
=(�
2
)
2
, therefore, �
2
,�
2
and �
2
are the primitive
Pythagorean triple. Written in Pythagorean form:
�
2
=2��,
�
2
=�
2
−�
2
,
�
2
=�
2
+�
2
,
where �,� are relatively prime (or co-prime) of opposite parity and �>�>0. Out of the three
equations above, we can rearrange �
2
=�
2
−�
2
into �
2
+�
2
=�
2
. Since �,� are relatively prime, we
have another set of primitive Pythagorean triple which are �,�,�. Additionally, �,� are also opposite in
parity and �>�>0, where � is odd and � is even and so the triple can be written out as follows:
�=2��,
�=�
2
−�
2
,
�=�
2
+�
2
,
whereby � and � are relatively prime of opposite parity and �>�>0. Henceforth we acquire the
following:
�
2
=2��=2(�
2
+�
2
)(2��)=4��(�
2
+�
2
),
�
2
2
2
=��(�
2
+�
2
).
This shows the ��(�
2
+�
2
) is the square of half of �, where � is an even integer (Edwards, 1977, Page
9 – 10).
2.3 Euclid’s Lemma and the proof of ??????=�
Euclid of Alexandria (325BC – 265BC) is a renowned mathematician known for his treatise, The Elements
(O’Connor and Robertson, 1999). Euclid’s Lemma states that if a prime number divides the product of
two numbers, it must divide at least one of these numbers but not both (Freeman, 2005).
Lemma. If any prime, ??????, divides ��. Then ?????? must divide either � or �.
Proof. Following the case of �=4, we know that ��(�
2
+�
2
) is a square. Since any prime, ??????, can
divide ��, we can say that �� and �
2
+�
2
are relatively prime since ?????? would have to divide � or � but
equation above can be written as (�
2
)
2
+(�
2
)
2
=(�
2
)
2
, therefore, �
2
,�
2
and �
2
are the primitive
Pythagorean triple. Written in Pythagorean form:
�
2
=2��,
�
2
=�
2
−�
2
,
�
2
=�
2
+�
2
,
where �,� are relatively prime (or co-prime) of opposite parity and �>�>0. Out of the three
equations above, we can rearrange �
2
=�
2
−�
2
into �
2
+�
2
=�
2
. Since �,� are relatively prime, we
have another set of primitive Pythagorean triple which are �,�,�. Additionally, �,� are also opposite in
parity and �>�>0, where � is odd and � is even and so the triple can be written out as follows:
�=2��,
�=�
2
−�
2
,
�=�
2
+�
2
,
whereby � and � are relatively prime of opposite parity and �>�>0. Henceforth we acquire the
following:
�
2
=2��=2(�
2
+�
2
)(2��)=4��(�
2
+�
2
),
�
2
2
2
=��(�
2
+�
2
).
This shows the ��(�
2
+�
2
) is the square of half of �, where � is an even integer (Edwards, 1977, Page
9 – 10).
2.3 Euclid’s Lemma and the proof of ??????=�
Euclid of Alexandria (325BC – 265BC) is a renowned mathematician known for his treatise, The Elements
(O’Connor and Robertson, 1999). Euclid’s Lemma states that if a prime number divides the product of
two numbers, it must divide at least one of these numbers but not both (Freeman, 2005).
Lemma. If any prime, ??????, divides ��. Then ?????? must divide either � or �.
Proof. Following the case of �=4, we know that ��(�
2
+�
2
) is a square. Since any prime, ??????, can
divide ��, we can say that �� and �
2
+�
2
are relatively prime since ?????? would have to divide � or � but
6
not both, and therefore �
2
+�
2
is not divisible by ??????. Now that we know �� is a square, � and � must
both be square since they are relatively prime. In other words, we can write both � and � as follows:
�=�
2
,
�=�
2
.
By substituting the above equation into �
2
+�
2
we get �
4
+�
4
. Since �
2
+�
2
is a square, �
4
+�
4
is
also a square, which leads to the fact that �
4
+�
4
=�
2
+�
2
=�<�
2
+�
2
=�<�
2
=�
4
+�
4
.
This begins a sequence of infinitely descending positive integers, which is implausible due to the same
argument as shown in section 2.1. Hence, the sum of two fourth power integers cannot equal to a
square (Edwards, 1977, Page 10).
Corollary. The equation �
4
+�
4
=�
4
has no integer solution where �,�,�≠0.
Proof. The equation above can be written as �
4
+�
4
=(�
2
)
2
. Subsequently the sum of two fourth
power integers cannot equal a square as shown above, therefore by contradiction, it cannot equal a
fourth power (Edwards, 1977, Page 10) thus proving the Last Theorem for �=4.
Due to this, we must now only prove that the theorem is true for � as an odd prime value. By proving
that the theorem is true for a given prime exponent, then it will be true for any exponent that is divisible
by the given prime value and in turn we will have shown the proof for all exponents (Freeman, 2005). To
explain this idea, supposed the equation �
�
+�
�
=�
�
has solutions when �≥2. Since � is greater
than or equal to 2, we can say that � is divisible by 4, hence �
??????
4,�
??????
4 and �
??????
4 are solutions to �
�
+�
�
=
�
�
. Similarly, if � is an odd prime (�), then �
??????
??????,�
??????
??????
and �
??????
??????
are also solutions to the equation. Therefore
by proving that no solutions exist for �=4 and �=�, then we have proven that there are no solutions
for any exponent (Mack – Crane, 2015).
not both, and therefore �
2
+�
2
is not divisible by ??????. Now that we know �� is a square, � and � must
both be square since they are relatively prime. In other words, we can write both � and � as follows:
�=�
2
,
�=�
2
.
By substituting the above equation into �
2
+�
2
we get �
4
+�
4
. Since �
2
+�
2
is a square, �
4
+�
4
is
also a square, which leads to the fact that �
4
+�
4
=�
2
+�
2
=�<�
2
+�
2
=�<�
2
=�
4
+�
4
.
This begins a sequence of infinitely descending positive integers, which is implausible due to the same
argument as shown in section 2.1. Hence, the sum of two fourth power integers cannot equal to a
square (Edwards, 1977, Page 10).
Corollary. The equation �
4
+�
4
=�
4
has no integer solution where �,�,�≠0.
Proof. The equation above can be written as �
4
+�
4
=(�
2
)
2
. Subsequently the sum of two fourth
power integers cannot equal a square as shown above, therefore by contradiction, it cannot equal a
fourth power (Edwards, 1977, Page 10) thus proving the Last Theorem for �=4.
Due to this, we must now only prove that the theorem is true for � as an odd prime value. By proving
that the theorem is true for a given prime exponent, then it will be true for any exponent that is divisible
by the given prime value and in turn we will have shown the proof for all exponents (Freeman, 2005). To
explain this idea, supposed the equation �
�
+�
�
=�
�
has solutions when �≥2. Since � is greater
than or equal to 2, we can say that � is divisible by 4, hence �
??????
4,�
??????
4 and �
??????
4 are solutions to �
�
+�
�
=
�
�
. Similarly, if � is an odd prime (�), then �
??????
??????,�
??????
??????
and �
??????
??????
are also solutions to the equation. Therefore
by proving that no solutions exist for �=4 and �=�, then we have proven that there are no solutions
for any exponent (Mack – Crane, 2015).
7
3 The case of ??????=�
Leonhard Euler (1707-1783) was a Swiss mathematician whom made important discoveries in
mathematics such as graph theory and number theory (Edwards, 1977, Page 39). Euler was also the first
mathematician to make a progress to Fermat’s Last Theorem with the proof for �=3, after Fermat,
who provided the proof for �=4 (Freeman, 2005).
Leonhard Euler had come up with two proofs for the case of �=3. His first proof had a mistake, which
he did not realise until trying to prove his lemma to the proof of �=3. In order to correct this, he gave
an alternative proof (Freeman, 2005). We will explore the details of this proof, which is greatly based on
the proofs shown by Harold M. Edwards in his book; Fermat’s Last Theorem: A Genetic introduction to
Algebraic Number Theory (1977) and by Larry Freeman in his blog; Fermat’s Last Theorem (2005).
Theorem 5. The equation �
3
+�
3
=�
3
has no positive integer solution �,�,�.
Euler applied Fermat’s method of infinite descent to show the proof to Fermat’s proposition; ‘A cube
cannot be equal to the sum of two non-zero cubes’ (Ribenboim, 2013, Page 24). Euler showed that if it
was possible to find positive integer solutions �,�,� for the equation �
3
+�
3
=�
3
, then one can find
smaller positive integer solutions to the equation. This would then make it possible to find even smaller
solutions, causing a continuous decreasing sequence of triples to the equation, which is impossible
(Edwards, 1977, Page 40).
Proof. In order to show this proof, as shown by both Larry Freeman and Harold M. Edwards, we must
first assume that �,�,�are co-prime. This means that the greatest common divisor of (�,�), (�,�) or
(�,�) is 1. Following a simpler notation, gcd(�,�,�)=1. Since �,�,� are co-prime, we know that at
least one of the integers is even. Therefore, if both � and � are odd, then � would be even since
(Edwards, 1977, Page 40);
���+���=����.
We also know that � and � are symmetrical, since the equation �
3
+�
3
=�
3
implies that both � and �
have similar properties, thus by showing the case of � being even, it will cover the case of � being even
(Freeman, 2005). Due to this, we can now split the proof into two cases, with case 1 showing if � is even
and case 2 showing if � is even.
3 The case of ??????=�
Leonhard Euler (1707-1783) was a Swiss mathematician whom made important discoveries in
mathematics such as graph theory and number theory (Edwards, 1977, Page 39). Euler was also the first
mathematician to make a progress to Fermat’s Last Theorem with the proof for �=3, after Fermat,
who provided the proof for �=4 (Freeman, 2005).
Leonhard Euler had come up with two proofs for the case of �=3. His first proof had a mistake, which
he did not realise until trying to prove his lemma to the proof of �=3. In order to correct this, he gave
an alternative proof (Freeman, 2005). We will explore the details of this proof, which is greatly based on
the proofs shown by Harold M. Edwards in his book; Fermat’s Last Theorem: A Genetic introduction to
Algebraic Number Theory (1977) and by Larry Freeman in his blog; Fermat’s Last Theorem (2005).
Theorem 5. The equation �
3
+�
3
=�
3
has no positive integer solution �,�,�.
Euler applied Fermat’s method of infinite descent to show the proof to Fermat’s proposition; ‘A cube
cannot be equal to the sum of two non-zero cubes’ (Ribenboim, 2013, Page 24). Euler showed that if it
was possible to find positive integer solutions �,�,� for the equation �
3
+�
3
=�
3
, then one can find
smaller positive integer solutions to the equation. This would then make it possible to find even smaller
solutions, causing a continuous decreasing sequence of triples to the equation, which is impossible
(Edwards, 1977, Page 40).
Proof. In order to show this proof, as shown by both Larry Freeman and Harold M. Edwards, we must
first assume that �,�,�are co-prime. This means that the greatest common divisor of (�,�), (�,�) or
(�,�) is 1. Following a simpler notation, gcd(�,�,�)=1. Since �,�,� are co-prime, we know that at
least one of the integers is even. Therefore, if both � and � are odd, then � would be even since
(Edwards, 1977, Page 40);
���+���=����.
We also know that � and � are symmetrical, since the equation �
3
+�
3
=�
3
implies that both � and �
have similar properties, thus by showing the case of � being even, it will cover the case of � being even
(Freeman, 2005). Due to this, we can now split the proof into two cases, with case 1 showing if � is even
and case 2 showing if � is even.
8
Case 1: ?????? is even
Lemma 2. If � is even, then �,� exists such that:
1. gcd(�,�)=1.
2. �,� are positive.
3. Either � or � is odd, therefore, have opposite parity.
4. �
3
+�
3
=�
3
only has a solution if �,� exists with the above properties.
Proof. Since we have assumed that �,�,� are co-prime and we have assumed that � is even for this case,
then � and � are odd such that (Edwards, 1977, Page 40 – 41):
�+�=����,
�−�=����.
With the above in mind, let:
2�=�+�,
2�=�−�.
Since any number �,� would be even if in the form 2�,2�. We will then have � and � such that:
�=
1
2
(2�+2�)=�+�,
�=
1
2
(2�−2�)=�−�.
This can be worked out by applying the method of solving simultaneous equations to 2�=�+� and
2�=�−�. We can now express �
3
+�
3
=�
3
in terms of �,�:
�
3
=�
3
+�
3
,
=(�+�)(�
2
−��+�
2
),
=(�+�+�−�)[(�+�)
2
−(�+�)(�−�)+(�−�)
2
],
=2�(�
2
+3�
2
).
Both � and � have opposite parity since both �+�=� and �−�=� are odd. Equally � and � also
have gcd(�,�)=1 since any factor that divides into � and � would divide into � and �, but that is
Case 1: ?????? is even
Lemma 2. If � is even, then �,� exists such that:
1. gcd(�,�)=1.
2. �,� are positive.
3. Either � or � is odd, therefore, have opposite parity.
4. �
3
+�
3
=�
3
only has a solution if �,� exists with the above properties.
Proof. Since we have assumed that �,�,� are co-prime and we have assumed that � is even for this case,
then � and � are odd such that (Edwards, 1977, Page 40 – 41):
�+�=����,
�−�=����.
With the above in mind, let:
2�=�+�,
2�=�−�.
Since any number �,� would be even if in the form 2�,2�. We will then have � and � such that:
�=
1
2
(2�+2�)=�+�,
�=
1
2
(2�−2�)=�−�.
This can be worked out by applying the method of solving simultaneous equations to 2�=�+� and
2�=�−�. We can now express �
3
+�
3
=�
3
in terms of �,�:
�
3
=�
3
+�
3
,
=(�+�)(�
2
−��+�
2
),
=(�+�+�−�)[(�+�)
2
−(�+�)(�−�)+(�−�)
2
],
=2�(�
2
+3�
2
).
Both � and � have opposite parity since both �+�=� and �−�=� are odd. Equally � and � also
have gcd(�,�)=1 since any factor that divides into � and � would divide into � and �, but that is
9
impossible as � and � are co-prime, hence it would be contradictory if that was the case. With this in
mind, it can be said that there can exist a solution �,� to �
3
+�
3
=�
3
due to the co-prime integers,
�,�, existing such that (Freeman, 2005):
�
3
=2�(�
2
+3�
2
)=����.
Case 2: ?????? is even
The same argument can be applied to case 2. In order to show the proof, we will be using Lemma 2 as
shown in case 1, but with the difference in being � is even rather than �.
Proof. As we know, �,�,� are co-prime, therefore, � and � are both odd. In turn �+� and �−� are
both even, therefore there exists �,� such that we let (Freeman, 2005):
2�=�−�,
2�=�+�,
due to the fact that the equation will be rearranged to �
3
=�
3
−�
3
since � is now even. Using
simultaneous equations, we have �,� such that:
�=�+�,
�=�−�.
Since �,� are both odd, � and � have opposite parity, and using the same argument as case 1,
gcd(�,�)=1. Therefore, there exists �,� such that:
�
3
=[�+�−(�−�)][(�+�)
2
+(�+�)(�−�)+(�−�)
2
]
=2�(�
2
+3�
2
)=����.
This results in the same conclusion as case 1 (Freeman, 2005).
Now that we have shown that regardless of either �,�,� being even, it will conclude with coprime
positive integers �,� such that 2�(�
2
+3�
2
) is a cube. The next step is to show that 2� and �
2
+3�
2
are co-prime and ‘that the only way that their product can be a cube is for each of them separately to be
a cube’ (Edwards, 1977, Page 41).
We know that � and � have opposite parity due to Lemma 2, therefore �
2
+3�
2
is odd, and we know
that � and � are co-prime, therefore we end up with the argument that if the above statement is true,
then gcd(2�,�
2
+3�
2
)=1 or 3.
impossible as � and � are co-prime, hence it would be contradictory if that was the case. With this in
mind, it can be said that there can exist a solution �,� to �
3
+�
3
=�
3
due to the co-prime integers,
�,�, existing such that (Freeman, 2005):
�
3
=2�(�
2
+3�
2
)=����.
Case 2: ?????? is even
The same argument can be applied to case 2. In order to show the proof, we will be using Lemma 2 as
shown in case 1, but with the difference in being � is even rather than �.
Proof. As we know, �,�,� are co-prime, therefore, � and � are both odd. In turn �+� and �−� are
both even, therefore there exists �,� such that we let (Freeman, 2005):
2�=�−�,
2�=�+�,
due to the fact that the equation will be rearranged to �
3
=�
3
−�
3
since � is now even. Using
simultaneous equations, we have �,� such that:
�=�+�,
�=�−�.
Since �,� are both odd, � and � have opposite parity, and using the same argument as case 1,
gcd(�,�)=1. Therefore, there exists �,� such that:
�
3
=[�+�−(�−�)][(�+�)
2
+(�+�)(�−�)+(�−�)
2
]
=2�(�
2
+3�
2
)=����.
This results in the same conclusion as case 1 (Freeman, 2005).
Now that we have shown that regardless of either �,�,� being even, it will conclude with coprime
positive integers �,� such that 2�(�
2
+3�
2
) is a cube. The next step is to show that 2� and �
2
+3�
2
are co-prime and ‘that the only way that their product can be a cube is for each of them separately to be
a cube’ (Edwards, 1977, Page 41).
We know that � and � have opposite parity due to Lemma 2, therefore �
2
+3�
2
is odd, and we know
that � and � are co-prime, therefore we end up with the argument that if the above statement is true,
then gcd(2�,�
2
+3�
2
)=1 or 3.
10
Lemma 3. If �,� are co-prime and have opposite parity, then gcd(2�,�
2
+3�
2
)=1��3.
Proof. We begin this proof by assuming that we have any positive integer �, such that � can divide into
2� and �
2
+3�
2
, therefore, � is a common factor of 2� and �
2
+3�
2
. We also know that � cannot be 2
since we have shown �
2
+3�
2
is odd, therefore we can assume that � is greater than 3. Consequently,
�,� exists such that (Freeman, 2005):
2�=��,
�
2
+3�
2
=��.
Even though we know � is not 2, we know that 2 can divide into � since 2�=��, and so there exists �
such that � is half of �, resulting in:
�=��.
We can now substitute the value of � into ��, and acquire the following:
�
2
+3�
2
=��,
3�
2
=��−�
2
,
3�
2
=��−(��)
2
,
=�(�−��
2
).
Due to the fact that � cannot divide into the 3 of 3�
2
, as it is greater than 3, it must divide into � since
we can apply Euclid’s Lemma as explained in section 2.3. However, this is contradictory since � is a
common factor of 2� and �
2
+3�
2
which, in turn, is a common factor of �,�
2
+3�
2
and �,3�
2
(Freeman, 2005).
Therefore, if � can divide into �, then it must divide into �, but this contradicts �,� being co-prime since
� divides into �. Therefore, we have two cases whereby we must argue that 3 does not divide into �,
gcd(2�,�
2
+3�
2
)=1, and that 3 does divide into �, gcd(2�,�
2
+3�
2
)=3 (Freeman, 2005).
Lemma 3. If �,� are co-prime and have opposite parity, then gcd(2�,�
2
+3�
2
)=1��3.
Proof. We begin this proof by assuming that we have any positive integer �, such that � can divide into
2� and �
2
+3�
2
, therefore, � is a common factor of 2� and �
2
+3�
2
. We also know that � cannot be 2
since we have shown �
2
+3�
2
is odd, therefore we can assume that � is greater than 3. Consequently,
�,� exists such that (Freeman, 2005):
2�=��,
�
2
+3�
2
=��.
Even though we know � is not 2, we know that 2 can divide into � since 2�=��, and so there exists �
such that � is half of �, resulting in:
�=��.
We can now substitute the value of � into ��, and acquire the following:
�
2
+3�
2
=��,
3�
2
=��−�
2
,
3�
2
=��−(��)
2
,
=�(�−��
2
).
Due to the fact that � cannot divide into the 3 of 3�
2
, as it is greater than 3, it must divide into � since
we can apply Euclid’s Lemma as explained in section 2.3. However, this is contradictory since � is a
common factor of 2� and �
2
+3�
2
which, in turn, is a common factor of �,�
2
+3�
2
and �,3�
2
(Freeman, 2005).
Therefore, if � can divide into �, then it must divide into �, but this contradicts �,� being co-prime since
� divides into �. Therefore, we have two cases whereby we must argue that 3 does not divide into �,
gcd(2�,�
2
+3�
2
)=1, and that 3 does divide into �, gcd(2�,�
2
+3�
2
)=3 (Freeman, 2005).
11
Case 1: 3 does not divide into ??????
Since we are taking up that case that gcd(2p,p
2
+3q
2
)=1, we can assume that 2p and p
2
+3q
2
are
cubes due to the Relatively Prime Divisors Lemma, the proof to this can be found in Larry Freeman’s
Blog; Fermat’s Last Theorem (2005). This Lemma states that if gcd(�,�)=1 and ��=�
�
, then there
exists �,� such that:
�=�
�
�=�
�
.
Applying this to our case, we know that �
3
=2�(�
2
+3�
2
) and gcd(2�,�
2
+3�
2
)=1, then there
exists �,� such that:
�
3
=2�
�
3
=�
2
+3�
2
.
There forth an assumption that 2�, �
2
+3�
2
are both cubes can be made.
Lemma 4. If gcd(2�,�
2
+3�
2
)=1, then there must be smaller solutions to Fermat’s Last Theorem: �=
3.
Proof. Since we know that 2�,�
2
+3�
2
are cube, then we know that there exists �,� such that
gcd(�,�)=1 and both �,� have opposite parities since �,� also have opposite parities. Let �
3
=�
2
+
3�
2
, since we are applying Relatively Prime Divisors Lemma (Freeman, 2005). We know that �,� have
opposite parities, in regards to which � can be stated as odd and so � will be in the form of �
2
+3�
2
because we know gcd(�,�)=1, which means that every odd factor in the form �
2
+3�
2
will have the
same form.
As a result of the above:
�
3
=(�
2
+3�
2
)
3
,
=�
2
+3�
2
(�
2
+3�
2
)
2
,
=(�
2
+3�
2
)(�
4
+6�
2
�
2
+9�
4
),
=(�
2
+3�
2
)(�
4
−6�
2
�
2
+12�
2
�
2
+9�
4
),
=(�
2
+3�
2
)[(�
2
−3�
2
)
2
+3(2�)
2
],
Case 1: 3 does not divide into ??????
Since we are taking up that case that gcd(2p,p
2
+3q
2
)=1, we can assume that 2p and p
2
+3q
2
are
cubes due to the Relatively Prime Divisors Lemma, the proof to this can be found in Larry Freeman’s
Blog; Fermat’s Last Theorem (2005). This Lemma states that if gcd(�,�)=1 and ��=�
�
, then there
exists �,� such that:
�=�
�
�=�
�
.
Applying this to our case, we know that �
3
=2�(�
2
+3�
2
) and gcd(2�,�
2
+3�
2
)=1, then there
exists �,� such that:
�
3
=2�
�
3
=�
2
+3�
2
.
There forth an assumption that 2�, �
2
+3�
2
are both cubes can be made.
Lemma 4. If gcd(2�,�
2
+3�
2
)=1, then there must be smaller solutions to Fermat’s Last Theorem: �=
3.
Proof. Since we know that 2�,�
2
+3�
2
are cube, then we know that there exists �,� such that
gcd(�,�)=1 and both �,� have opposite parities since �,� also have opposite parities. Let �
3
=�
2
+
3�
2
, since we are applying Relatively Prime Divisors Lemma (Freeman, 2005). We know that �,� have
opposite parities, in regards to which � can be stated as odd and so � will be in the form of �
2
+3�
2
because we know gcd(�,�)=1, which means that every odd factor in the form �
2
+3�
2
will have the
same form.
As a result of the above:
�
3
=(�
2
+3�
2
)
3
,
=�
2
+3�
2
(�
2
+3�
2
)
2
,
=(�
2
+3�
2
)(�
4
+6�
2
�
2
+9�
4
),
=(�
2
+3�
2
)(�
4
−6�
2
�
2
+12�
2
�
2
+9�
4
),
=(�
2
+3�
2
)[(�
2
−3�
2
)
2
+3(2�)
2
],
12
=[�(�
2
−3�
2
)−3�(2��)]
2
+3[�(2��)+�(�
2
−3�
2
)]
2
,
=(�
3
−3��
2
−6��
2
)
2
+3(2�
2
�+�
2
�−3�
3
)
2
,
=(�
3
−9��
2
)
2
+3(3�
2
�−3�
3
)
2
.
As �
3
=�
2
+3�
2
, then �
2
+3�
2
=(�
3
−9��
2
)
2
+3(3�
2
�−3�
3
)
2
. This allows us to define �,�
such that :
�=�
3
−9��
2
=�(�−3�)(�+3�),
�=3�
2
�−3�
3
=3�(�−�)(�+�),
where gcd(�,�)=1.
We can now substitute the value for � into 2�, therefore (Freeman, 2005):
2�=2(�
3
−9��
2
),
=2�
3
−18��
2
,
=2�(�−3�)(�+3�).
Since �,� are opposite parities, we know that both �−3� and �+3� are odd. Due to this, any
common factor that divides into 2�,�±3� would also be a common factor of �,�±3� and
subsequently, be a common factor to �,±3� (Edwards, 1977, Page 42). We know that this common
factor cannot be 2 since 2 cannot divide into �±3�, therefore the only common factor would be 3. But
if 3 divides into �, it would then be able to divide into � which is impossible since gcd(2�,�
2
+3�
2
)=
1.
As a result of this, 2�,�±3� are co-prime and all three are cube due to the Relative Prime Divisors
Lemma. We now have �,�,� such that (Edwards, 1977, Page 42):
�
3
=2�
�
3
=�−3�
�
3
=�+3�.
From the above equations, we can see that �
3
+�
3
=2�=�
3
, this gives a solution smaller than �,�,�
where �
3
+�
3
=�
3
. We know this is the case since (�
3
)(�
3
)(�
3
)=2�(�−3�)(�+3�)=2�,
where �
3
or �
3
is 2�(�
2
+3�
2
) as shown in case 1: ?????? is even and case 2: ?????? is even.
=[�(�
2
−3�
2
)−3�(2��)]
2
+3[�(2��)+�(�
2
−3�
2
)]
2
,
=(�
3
−3��
2
−6��
2
)
2
+3(2�
2
�+�
2
�−3�
3
)
2
,
=(�
3
−9��
2
)
2
+3(3�
2
�−3�
3
)
2
.
As �
3
=�
2
+3�
2
, then �
2
+3�
2
=(�
3
−9��
2
)
2
+3(3�
2
�−3�
3
)
2
. This allows us to define �,�
such that :
�=�
3
−9��
2
=�(�−3�)(�+3�),
�=3�
2
�−3�
3
=3�(�−�)(�+�),
where gcd(�,�)=1.
We can now substitute the value for � into 2�, therefore (Freeman, 2005):
2�=2(�
3
−9��
2
),
=2�
3
−18��
2
,
=2�(�−3�)(�+3�).
Since �,� are opposite parities, we know that both �−3� and �+3� are odd. Due to this, any
common factor that divides into 2�,�±3� would also be a common factor of �,�±3� and
subsequently, be a common factor to �,±3� (Edwards, 1977, Page 42). We know that this common
factor cannot be 2 since 2 cannot divide into �±3�, therefore the only common factor would be 3. But
if 3 divides into �, it would then be able to divide into � which is impossible since gcd(2�,�
2
+3�
2
)=
1.
As a result of this, 2�,�±3� are co-prime and all three are cube due to the Relative Prime Divisors
Lemma. We now have �,�,� such that (Edwards, 1977, Page 42):
�
3
=2�
�
3
=�−3�
�
3
=�+3�.
From the above equations, we can see that �
3
+�
3
=2�=�
3
, this gives a solution smaller than �,�,�
where �
3
+�
3
=�
3
. We know this is the case since (�
3
)(�
3
)(�
3
)=2�(�−3�)(�+3�)=2�,
where �
3
or �
3
is 2�(�
2
+3�
2
) as shown in case 1: ?????? is even and case 2: ?????? is even.
13
Case 2: 3 divides into ??????
Now, our final case to show that �
3
+�
3
=�
3
has no integer solution �,�,� only differs slightly from
the previous case. In both cases, we have the same assumptions, therefore, �,� are co-prime with
opposite parity and 2�,�
2
+3�
2
are a cube. The difference, in this case, would be the fact that
gcd(2�,�
2
+3�
2
)=3 (Freeman, 2005).
Lemma 5. If gcd(2�,�
2
+3�
2
)=3, then there must be smaller solutions to Fermat’s Last Theorem:
�=3.
Proof. To begin, we note that 3 divides into � but does not divide into �, since 3 can divide into 2� as
the gcd(2�,�
2
+3�
2
)=3 (Edwards, 1977, Page 42). But � and � are co-prime, therefore due to
Euclid’s Lemma, if 3 divides into �, it cannot divide into �. Since 3 divides into �, there exists � such that
(Freeman, 2005):
�=3�.
By substituting the value of � into 2�(�
2
+3�
2
) we acquire:
2�(�
2
+3�
2
)=[2(3�)((3�)
2
+3�
2
)],
=3
2
(2�)(3�
2
+�
2
).
Since 3 cannot divide �, it cannot divide 3�
2
+�
2
. As a result of this, 3
2
(2�) and 3�
2
+�
2
are co-
prime. Consequently, � and � are also of opposite parity, and therefore gcd(�,�)=1 since gcd(�,�)=
1. We know that 2�(�
2
+3�
2
) is cube, thus 3
2
(2�),3�
2
+�
2
are also cubes, since 2�(�
2
+3�
2
)=
3
2
(2�)(3�
2
+�
2
).
From the previous case, we calculated �
2
+3�
2
=(�
3
−9��
2
)
2
+3(3�
2
�−3�
3
)
2
, therefore, as
before, there exists �,� such that (Freeman, 2005):
�=�
3
−9��
2
=�(�−3�)(�+3�),
�=3�
2
�−3�
3
=3�(�−�)(�+�),
where gcd(�,�)=1.
Due to the fact that we know 3
2
(2�) is a cube, we can define �,� further such that:
3
2
(2�)=3
2
[2(3�
2
�−3�
3
)],
Case 2: 3 divides into ??????
Now, our final case to show that �
3
+�
3
=�
3
has no integer solution �,�,� only differs slightly from
the previous case. In both cases, we have the same assumptions, therefore, �,� are co-prime with
opposite parity and 2�,�
2
+3�
2
are a cube. The difference, in this case, would be the fact that
gcd(2�,�
2
+3�
2
)=3 (Freeman, 2005).
Lemma 5. If gcd(2�,�
2
+3�
2
)=3, then there must be smaller solutions to Fermat’s Last Theorem:
�=3.
Proof. To begin, we note that 3 divides into � but does not divide into �, since 3 can divide into 2� as
the gcd(2�,�
2
+3�
2
)=3 (Edwards, 1977, Page 42). But � and � are co-prime, therefore due to
Euclid’s Lemma, if 3 divides into �, it cannot divide into �. Since 3 divides into �, there exists � such that
(Freeman, 2005):
�=3�.
By substituting the value of � into 2�(�
2
+3�
2
) we acquire:
2�(�
2
+3�
2
)=[2(3�)((3�)
2
+3�
2
)],
=3
2
(2�)(3�
2
+�
2
).
Since 3 cannot divide �, it cannot divide 3�
2
+�
2
. As a result of this, 3
2
(2�) and 3�
2
+�
2
are co-
prime. Consequently, � and � are also of opposite parity, and therefore gcd(�,�)=1 since gcd(�,�)=
1. We know that 2�(�
2
+3�
2
) is cube, thus 3
2
(2�),3�
2
+�
2
are also cubes, since 2�(�
2
+3�
2
)=
3
2
(2�)(3�
2
+�
2
).
From the previous case, we calculated �
2
+3�
2
=(�
3
−9��
2
)
2
+3(3�
2
�−3�
3
)
2
, therefore, as
before, there exists �,� such that (Freeman, 2005):
�=�
3
−9��
2
=�(�−3�)(�+3�),
�=3�
2
�−3�
3
=3�(�−�)(�+�),
where gcd(�,�)=1.
Due to the fact that we know 3
2
(2�) is a cube, we can define �,� further such that:
3
2
(2�)=3
2
[2(3�
2
�−3�
3
)],
14
=3
3
[2(�
2
�−�
3
)],
=3
3
[2�(�
2
−�
2
)],
=3
3
[2�(�−�)(�+�)].
Following from the fact that 3
2
(2�) is a cube, then 3
3
[2�(�−�)(�+�)] is a cube, and subsequently
2�(�−�)(�+�) is also a cube. For that reason, there exists �,�,� such that:
�
3
=2�
�
3
=�−�
�
3
=�+�.
This, in turn, gives another solution to Fermat’s Last Theorem; �=3 as �
3
+�
3
=2�=�
3
, whereby
�
3
�
3
�
3
is less than �=3�
2
�−3�
3
=3�(�−�)(�+�) which is also less than �=3�, which is less
than �
3
or �
3
=2�(�
2
+3�
2
) as well (Edwards, 1977, Page 42).
Corollary 2. �
�
+�
3
=�
3
has no integer solution �,�,�.
Proof. In both cases where gcd(2�,�
2
+3�
2
)=1 or 3, we acquire smaller and smaller solutions to the
already existing small solutions, which is impossible. Therefore, this proves that there are no integer
solutions to the equation by the method of infinite descent.
=3
3
[2(�
2
�−�
3
)],
=3
3
[2�(�
2
−�
2
)],
=3
3
[2�(�−�)(�+�)].
Following from the fact that 3
2
(2�) is a cube, then 3
3
[2�(�−�)(�+�)] is a cube, and subsequently
2�(�−�)(�+�) is also a cube. For that reason, there exists �,�,� such that:
�
3
=2�
�
3
=�−�
�
3
=�+�.
This, in turn, gives another solution to Fermat’s Last Theorem; �=3 as �
3
+�
3
=2�=�
3
, whereby
�
3
�
3
�
3
is less than �=3�
2
�−3�
3
=3�(�−�)(�+�) which is also less than �=3�, which is less
than �
3
or �
3
=2�(�
2
+3�
2
) as well (Edwards, 1977, Page 42).
Corollary 2. �
�
+�
3
=�
3
has no integer solution �,�,�.
Proof. In both cases where gcd(2�,�
2
+3�
2
)=1 or 3, we acquire smaller and smaller solutions to the
already existing small solutions, which is impossible. Therefore, this proves that there are no integer
solutions to the equation by the method of infinite descent.
15
4 Ernst Kummer and the proof for regular primes
Ernst Kummer’s (January 29
th
1810 – May 14
th
1893) contribution to the proof of the Last Theorem was
of great extent, aided by Gabriel Lamé (July 22
nd
1795 – May 1
st
1870) failed proof, Kummer was able to
finalise his proof for regular primes (Freeman, 2006). Kummer was a 19
th
century German
mathematician whom made substantial progress to FLT after Leonhard Euler (Varma, 2008). Many
mathematicians have tried to prove the theorem with many special cases such as the proof of �=5 by
Adrien-Marie Legendre (September 18
th
1752 – January 10
th
1833) and Peter Gustav Lejeune Dirichlet
(February 13
th
1805 – May 5
th
1859), by using Sophie Germain's (April 1
st
1776 – June 27
th
1831)
Theorem, and the proof of �=7 by Lamé that was later replaced by Kummer's proof for regular primes
(Edwards, 1977, Page 59 – 60).
Kummer came across Fermat’s Last Theorem whilst generalising the law of quadratic reciprocity (Varma,
2008). He himself stated that he had no interest in the Last Theorem (Freeman, 2006), but rather
considered it to be a ‘curiosity of number theory since it was a problem that could be understood by a
12 year old’ (Varma, 2008). The proofs that will be discussed in the latter paragraphs will be largely
based on the proofs shown by Harold M. Edwards in his book; Fermat’s Last Theorem: A Genetic
Introduction to Algebraic Number Theory (1977), Larry Freeman in his blog; Fermat’s Last Theorem
(2005), Ila Varma in her article; Kummer, Regular Primes, and Fermat’s Last Theorem (2008) and by
Sander Mack – Crane in his report; Fermat’s Last Theorem for Regular Primes (2015).
4.1 The Cyclotomic ring/field and Root of unity
As explained in section 2.3, in order to acquire a proof for the general case, we will need to prove that
no solution exists when � is any odd prime (Freeman, 2005). Before we begin with the proof for regular
primes, we must first define some facts about cyclotomic rings and algebraic number fields. After this,
we will be able to understand the core idea of the proof (Varma, 2008).
In order for these technical terms to be understood by everyone, the definition of rings and fields will be
explained as such; a ring is a set of numbers that can be added, subtracted and multiplied whilst within
the set. Subsequently, a field is a set of numbers that can be added, subtracted, multiplied and divided
whilst within the set. Rings will be defined by notation ℤ, whereas fields will use the notation ℚ. We can
see that in the definition for rings and fields, both differ by the fact that one is able to divide within the
set whilst the other does not, however this is not the case. We can say that � divides �, or write �|� if
there exists an integer, �, in the ring, such that �=��. Therefore in terms of the integer set ℤ, 3 divides
6 since there is an integer �=2 which results these numbers being in the ring and the set. Conversely,
5|6 cannot exist in the set as there is no value for the integer �; 6=5×� (Mack – Crane, 2015).
4 Ernst Kummer and the proof for regular primes
Ernst Kummer’s (January 29
th
1810 – May 14
th
1893) contribution to the proof of the Last Theorem was
of great extent, aided by Gabriel Lamé (July 22
nd
1795 – May 1
st
1870) failed proof, Kummer was able to
finalise his proof for regular primes (Freeman, 2006). Kummer was a 19
th
century German
mathematician whom made substantial progress to FLT after Leonhard Euler (Varma, 2008). Many
mathematicians have tried to prove the theorem with many special cases such as the proof of �=5 by
Adrien-Marie Legendre (September 18
th
1752 – January 10
th
1833) and Peter Gustav Lejeune Dirichlet
(February 13
th
1805 – May 5
th
1859), by using Sophie Germain's (April 1
st
1776 – June 27
th
1831)
Theorem, and the proof of �=7 by Lamé that was later replaced by Kummer's proof for regular primes
(Edwards, 1977, Page 59 – 60).
Kummer came across Fermat’s Last Theorem whilst generalising the law of quadratic reciprocity (Varma,
2008). He himself stated that he had no interest in the Last Theorem (Freeman, 2006), but rather
considered it to be a ‘curiosity of number theory since it was a problem that could be understood by a
12 year old’ (Varma, 2008). The proofs that will be discussed in the latter paragraphs will be largely
based on the proofs shown by Harold M. Edwards in his book; Fermat’s Last Theorem: A Genetic
Introduction to Algebraic Number Theory (1977), Larry Freeman in his blog; Fermat’s Last Theorem
(2005), Ila Varma in her article; Kummer, Regular Primes, and Fermat’s Last Theorem (2008) and by
Sander Mack – Crane in his report; Fermat’s Last Theorem for Regular Primes (2015).
4.1 The Cyclotomic ring/field and Root of unity
As explained in section 2.3, in order to acquire a proof for the general case, we will need to prove that
no solution exists when � is any odd prime (Freeman, 2005). Before we begin with the proof for regular
primes, we must first define some facts about cyclotomic rings and algebraic number fields. After this,
we will be able to understand the core idea of the proof (Varma, 2008).
In order for these technical terms to be understood by everyone, the definition of rings and fields will be
explained as such; a ring is a set of numbers that can be added, subtracted and multiplied whilst within
the set. Subsequently, a field is a set of numbers that can be added, subtracted, multiplied and divided
whilst within the set. Rings will be defined by notation ℤ, whereas fields will use the notation ℚ. We can
see that in the definition for rings and fields, both differ by the fact that one is able to divide within the
set whilst the other does not, however this is not the case. We can say that � divides �, or write �|� if
there exists an integer, �, in the ring, such that �=��. Therefore in terms of the integer set ℤ, 3 divides
6 since there is an integer �=2 which results these numbers being in the ring and the set. Conversely,
5|6 cannot exist in the set as there is no value for the integer �; 6=5×� (Mack – Crane, 2015).
16
This is not the case with fields, since all divisions can occur in the set ℚ. An example of this is as follows;
5|6 can exist in the set due to the fact that the rational umber 5 can divide into 6 such that there exists
integer �=
6
5
; 6=5×
6
5
. From this it can be shown that every number in the field, ℚ, can become
irreducible, given that we are able to factorize a number into smaller numbers until it cannot factor
anymore. Henceforth a number can become a product of a prime (Mack – Crane, 2015).
This can now be followed up with the explanation of algebraic number fields (we will refer to this as
number fields), which is acquired from the addition of rational numbers to a root of a polynomial.
Consider the polynomial �
�
−1; this polynomial has a complex number ζ such that ??????
??????
=1, this is
known as the primitive �
??????ℎ
root of unity (Varma, 2008) as we allow ??????
??????
as the primitive root to the above
polynomial. Since we are proving for regular primes, let the primitive root be ??????
??????
??????
. By the addition of this
complex number to the field ℚ, we acquire the number field ℚ(??????
??????) that is written in form (Edwards,
1977, Page 80-81):
�
0+�
1??????
??????+�
2??????
??????
2
+⋯+�
??????−1??????
??????
??????−1
.
We are able to write the above number field in the form shown, due to the fact that each element of
the field is a complex number that has the form �+�?????? (all complex numbers have this form), where �,�
are the rational numbers and ?????? being the imaginary unit that represents ??????
?????? in our case.
Kummer had used λ to denote prime numbers and α to represent the imaginary unit such that ??????
??????
=1 is
the primitive ??????
??????ℎ
root of unity, as opposed to the notations we have used (Freeman, 2006). Regardless
of which notation is used, Kummer regarded these numbers as a ‘special type’ of complex numbers.
These are now known as the cyclotomic integers, due to way ζ is defined as a point on “the circle, |�|=
1 of a complex � – plane” that effects the way the circle is divided into � equal parts. The ‘cycl-’ part of
cyclotomic means circle and ‘tom-’ means divide (Edwards, 1977, Page 81).
4.2 Gabriel Lamé’s Proposed Proof
Although Kummer did not take interest into Fermat’s Last Theorem, he did make substantial steps into
proving the theorem (Freeman, 2006). This came merely a few weeks after Lamé claimed to have found
the complete proof to the theorem, which was announced in 1847 at the Paris Academy (Edwards,
1977, Page 76). Though his evidence was flawed, the concept behind it was simple and helped to
develop the general proof to the theorem. Having worked on other cases, such as the case for �=7;
Lamé noticed a trend whereby the difficulty to find the solutions increased as the value for � increased
(Edwards, 1977, Page 76). As seen from previous chapters, in order to find the proof for each value of �,
we must factorise the equation. In the search to simplify these factorizations of high � values, he
This is not the case with fields, since all divisions can occur in the set ℚ. An example of this is as follows;
5|6 can exist in the set due to the fact that the rational umber 5 can divide into 6 such that there exists
integer �=
6
5
; 6=5×
6
5
. From this it can be shown that every number in the field, ℚ, can become
irreducible, given that we are able to factorize a number into smaller numbers until it cannot factor
anymore. Henceforth a number can become a product of a prime (Mack – Crane, 2015).
This can now be followed up with the explanation of algebraic number fields (we will refer to this as
number fields), which is acquired from the addition of rational numbers to a root of a polynomial.
Consider the polynomial �
�
−1; this polynomial has a complex number ζ such that ??????
??????
=1, this is
known as the primitive �
??????ℎ
root of unity (Varma, 2008) as we allow ??????
??????
as the primitive root to the above
polynomial. Since we are proving for regular primes, let the primitive root be ??????
??????
??????
. By the addition of this
complex number to the field ℚ, we acquire the number field ℚ(??????
??????) that is written in form (Edwards,
1977, Page 80-81):
�
0+�
1??????
??????+�
2??????
??????
2
+⋯+�
??????−1??????
??????
??????−1
.
We are able to write the above number field in the form shown, due to the fact that each element of
the field is a complex number that has the form �+�?????? (all complex numbers have this form), where �,�
are the rational numbers and ?????? being the imaginary unit that represents ??????
?????? in our case.
Kummer had used λ to denote prime numbers and α to represent the imaginary unit such that ??????
??????
=1 is
the primitive ??????
??????ℎ
root of unity, as opposed to the notations we have used (Freeman, 2006). Regardless
of which notation is used, Kummer regarded these numbers as a ‘special type’ of complex numbers.
These are now known as the cyclotomic integers, due to way ζ is defined as a point on “the circle, |�|=
1 of a complex � – plane” that effects the way the circle is divided into � equal parts. The ‘cycl-’ part of
cyclotomic means circle and ‘tom-’ means divide (Edwards, 1977, Page 81).
4.2 Gabriel Lamé’s Proposed Proof
Although Kummer did not take interest into Fermat’s Last Theorem, he did make substantial steps into
proving the theorem (Freeman, 2006). This came merely a few weeks after Lamé claimed to have found
the complete proof to the theorem, which was announced in 1847 at the Paris Academy (Edwards,
1977, Page 76). Though his evidence was flawed, the concept behind it was simple and helped to
develop the general proof to the theorem. Having worked on other cases, such as the case for �=7;
Lamé noticed a trend whereby the difficulty to find the solutions increased as the value for � increased
(Edwards, 1977, Page 76). As seen from previous chapters, in order to find the proof for each value of �,
we must factorise the equation. In the search to simplify these factorizations of high � values, he
17
realised ‘that this can be overcome by decomposing �
�
+�
�
into � linear factors’ (Edwards, 1977, Page
76). This was made possible by Lamé by introducing the complex number, ??????
??????, such that ??????
??????
??????
=1 as shown
in section 4.1 (Varma, 2008). By using the algebraic identity with the root of unity gives way to the
following:
�
??????
+�
??????
=(�+�)(�+??????
??????�)(�+??????
??????
2
�)…(�+??????
??????
??????−1
�).
Lamé had applied techniques of factorizing with complex numbers, which have been used before in
Fermat’s Last Theorem, but no one had claimed to have used it until Lamé (Freeman, 2006). However,
though he claimed to have found the proof, he did not claim the entire credit for himself since the idea
was suggested to him by his colleague Joseph Liouville (March 24
th
1809 – September 8
th
1882)
(Edwards, 1977, Page 77).
Explained in a more understanding way, Lamé tried to show that if each factor �+�,�+??????
??????�, … , �+
??????
??????
??????−1
� are relatively prime then the equation �
??????
+�
??????
=�
??????
tells us that each said factor must be the
�
??????ℎ
power since each factor is equal to �
??????
, following this an infinite descent must be derived in order to
finalise the proof which is an impossible task. However, if all factors mentioned are not relatively prime,
then Lamé tried to show that all factors have a common factor �, such that
(�+�)
�
,
(�+????????????)
�
, … ,
(�+??????
??????
??????−1
�)
�
are relatively prime, which again has the impossible task to derive an infinite descent (Edwards, 1977,
page 77).
Lamé had thought that this was the revolutionary step needed to generate a complete solution, but this
was far from the case since many mathematicians, including Liouville, criticised Lamé’s announcement.
The main argument begin that the unique factorisation did not hold in the cyclotomic field ℚ(??????
??????)
(Varma, 2008), even though each element of the cyclotomic ring ℤ(??????
??????) could be factorized into a
product of a prime (Mack - Crane, 2015). More specifically, the unique factorization did not hold for all
primes �≥23. This was proven by Kummer three years before Lamé made his claim (Freeman, 2006),
however he had published it in a ‘very obscure’ place, which unknown to many mathematicians at the
time (Edwards, 1977, Page 78).
4.3 Ideal Numbers
From Lamé’s failed proof, a new problem arised; if these cyclotomic integers, that Lamé formed for his
‘proof’, did not have the unique factorization, then we must find ‘ideal numbers’ that fitted the
properties needed to restore the unique factorization. To begin with, the basic property that these ‘ideal
numbers’ need is its divisibility. Expanding on this point, if an ideal number is able to divide an element
in the cyclotomic ring, then it should be able to divide the product of two elements in the ring. In a more
realised ‘that this can be overcome by decomposing �
�
+�
�
into � linear factors’ (Edwards, 1977, Page
76). This was made possible by Lamé by introducing the complex number, ??????
??????, such that ??????
??????
??????
=1 as shown
in section 4.1 (Varma, 2008). By using the algebraic identity with the root of unity gives way to the
following:
�
??????
+�
??????
=(�+�)(�+??????
??????�)(�+??????
??????
2
�)…(�+??????
??????
??????−1
�).
Lamé had applied techniques of factorizing with complex numbers, which have been used before in
Fermat’s Last Theorem, but no one had claimed to have used it until Lamé (Freeman, 2006). However,
though he claimed to have found the proof, he did not claim the entire credit for himself since the idea
was suggested to him by his colleague Joseph Liouville (March 24
th
1809 – September 8
th
1882)
(Edwards, 1977, Page 77).
Explained in a more understanding way, Lamé tried to show that if each factor �+�,�+??????
??????�, … , �+
??????
??????
??????−1
� are relatively prime then the equation �
??????
+�
??????
=�
??????
tells us that each said factor must be the
�
??????ℎ
power since each factor is equal to �
??????
, following this an infinite descent must be derived in order to
finalise the proof which is an impossible task. However, if all factors mentioned are not relatively prime,
then Lamé tried to show that all factors have a common factor �, such that
(�+�)
�
,
(�+????????????)
�
, … ,
(�+??????
??????
??????−1
�)
�
are relatively prime, which again has the impossible task to derive an infinite descent (Edwards, 1977,
page 77).
Lamé had thought that this was the revolutionary step needed to generate a complete solution, but this
was far from the case since many mathematicians, including Liouville, criticised Lamé’s announcement.
The main argument begin that the unique factorisation did not hold in the cyclotomic field ℚ(??????
??????)
(Varma, 2008), even though each element of the cyclotomic ring ℤ(??????
??????) could be factorized into a
product of a prime (Mack - Crane, 2015). More specifically, the unique factorization did not hold for all
primes �≥23. This was proven by Kummer three years before Lamé made his claim (Freeman, 2006),
however he had published it in a ‘very obscure’ place, which unknown to many mathematicians at the
time (Edwards, 1977, Page 78).
4.3 Ideal Numbers
From Lamé’s failed proof, a new problem arised; if these cyclotomic integers, that Lamé formed for his
‘proof’, did not have the unique factorization, then we must find ‘ideal numbers’ that fitted the
properties needed to restore the unique factorization. To begin with, the basic property that these ‘ideal
numbers’ need is its divisibility. Expanding on this point, if an ideal number is able to divide an element
in the cyclotomic ring, then it should be able to divide the product of two elements in the ring. In a more
18
visual form, if an ideal number divides � in a cyclotomic ring, then it can divide ��. Subsequently, if it
can divide � and �, then it can divide �+� (Mack – Crane, 2015).
With the above property in mind, an ‘ideal number’ can be described by showing the many collections
of numbers it can divide. Henceforth, we can use this description to explain an ideal prime number; we
let ideal number � to be prime, if there exists �� in the cyclotomic ring, such that if �� is in � (therefore
allowing ideal � in its own ring to be a subset of the cyclotomic ring) then � or � is in �. This allows � to
be a cyclotomic integers, otherwise the ideal number will not exist as a cyclotomic integer without the
property shown above. By multiplying ideal numbers, we can let the product of these numbers be the
smallest kind that contains the cyclotomic integers, thus we can let �� to be the smallest ideals
containing the product of cyclotomic integers, such as �� (Mack – Crane, 2015).
Let us consider a set of ideal numbers in a cyclotomic ring (therefore these numbers are a subset of the
ring), we can say that each element (α) of the ring can produce an ideal of the element, which can be
divided by the said element (α). This element/ divisor is known as a principle (Edwards, 1977, Page 162).
By introducing these principle ideal numbers, the unique factorization is restored, since ‘every ideal in
the ring of integers can be written uniquely as a product of prime ideals’ (Mack – Crane, 2015).
To help aid us with the above idea, consider the example used in the definitions of rings in section 4.1.
We have the equation 6=2×3, now if we consider the ring ℤ(√−5) with the field ℚ(√−5) then we
can write the equation as (Varma, 2008):
6=2∗3=(1+√−5)∗(1−√−5),
whereby 2,3,(1+√−5),(1−√−5) are irreducible elements to ℤ(√−5). However, with the
introduction of ideal numbers, the factors shown above can now be further factorized such that (Mack –
Crane, 2015):
2=(2,1+√−5)(2,1−√−5)=�
1,�
2,
3=(3,1+√−5)(3,1−√−5)=�
3,�
4,
1+√−5=(2,1+√−5)(3,1+√−5)=�
1,�
3,
1−√−5=(2,1−√−5)(3,1−√−5)=�
2,�
4.
We have now further factorized the equation into ideal prime factors �
1,�
2,�
3,�
4 that are unique and
hold in the cyclotomic field ℚ(√−5). As shown above, the unique factorization does not always hold in
visual form, if an ideal number divides � in a cyclotomic ring, then it can divide ��. Subsequently, if it
can divide � and �, then it can divide �+� (Mack – Crane, 2015).
With the above property in mind, an ‘ideal number’ can be described by showing the many collections
of numbers it can divide. Henceforth, we can use this description to explain an ideal prime number; we
let ideal number � to be prime, if there exists �� in the cyclotomic ring, such that if �� is in � (therefore
allowing ideal � in its own ring to be a subset of the cyclotomic ring) then � or � is in �. This allows � to
be a cyclotomic integers, otherwise the ideal number will not exist as a cyclotomic integer without the
property shown above. By multiplying ideal numbers, we can let the product of these numbers be the
smallest kind that contains the cyclotomic integers, thus we can let �� to be the smallest ideals
containing the product of cyclotomic integers, such as �� (Mack – Crane, 2015).
Let us consider a set of ideal numbers in a cyclotomic ring (therefore these numbers are a subset of the
ring), we can say that each element (α) of the ring can produce an ideal of the element, which can be
divided by the said element (α). This element/ divisor is known as a principle (Edwards, 1977, Page 162).
By introducing these principle ideal numbers, the unique factorization is restored, since ‘every ideal in
the ring of integers can be written uniquely as a product of prime ideals’ (Mack – Crane, 2015).
To help aid us with the above idea, consider the example used in the definitions of rings in section 4.1.
We have the equation 6=2×3, now if we consider the ring ℤ(√−5) with the field ℚ(√−5) then we
can write the equation as (Varma, 2008):
6=2∗3=(1+√−5)∗(1−√−5),
whereby 2,3,(1+√−5),(1−√−5) are irreducible elements to ℤ(√−5). However, with the
introduction of ideal numbers, the factors shown above can now be further factorized such that (Mack –
Crane, 2015):
2=(2,1+√−5)(2,1−√−5)=�
1,�
2,
3=(3,1+√−5)(3,1−√−5)=�
3,�
4,
1+√−5=(2,1+√−5)(3,1+√−5)=�
1,�
3,
1−√−5=(2,1−√−5)(3,1−√−5)=�
2,�
4.
We have now further factorized the equation into ideal prime factors �
1,�
2,�
3,�
4 that are unique and
hold in the cyclotomic field ℚ(√−5). As shown above, the unique factorization does not always hold in
19
the original ring, however after the addition of principle ideal numbers the unique factorization holds,
therefore it will hold for the cyclotomic integers that created the ideal numbers. From this, we can
define class numbers. Class numbers are used to measure how far a ring is from unique factorization,
since elements from cyclotomic rings can produce principle ideal numbers, class numbers tells us the
number of ideals added into order to restore the unique factorization (Edwards, 1977, Page 163).
4.4 The proof for Regular Primes
In this final chapter that concludes the proof for regular primes, we will explore the mathematics
involved in solving this case, but we will not go into the full details of the proof. This chapter will give an
idea on how to solve Fermat’s Last Theorem for�=���������??????��(�).
Where Lamé failed to create a general proof, he did however succeed in establishing Fermat’s Last
Theorem. Following from the concept of class numbers, there are still a few odd primes that have class
numbers. These numbers, in the field ℚ(??????
??????) are 3,5,7,11,13,17,19. To salvage this and solve the final
part of the proof we turn to regular primes. We call any prime, � a regular prime if and only if it does not
divide the class number in the field ℚ(??????
??????). By solving for regular primes, we are able to complete the
proof for the theorem (Mack – Crane, 2015).
Theorem 6. The equation �
??????
+�
??????
=�
??????
has no integer solution �,�,� if � is any regular prime.
In order to show the overview of the proof, we will split the theorem into two cases, as shown for the
case of �=3 in chapter three. Whereby both cases show if any of the integers �,�,� are divisible by �.
Case 1: not divisible by ??????
Lemma 6. The integers �,�,� are co-prime and not divisible by �
Proof overview. We can write the equation in Theorem 6. as (Edwards, 1977, Page 168 – 173):
�
??????
=�
??????
+�
??????
=(�+�)(�+??????
??????�)(�+??????
??????
2
�)…(�+??????
??????
??????−1
�).
The first part of this proof is to show that all factors from the above equation are all co-prime. Due to
the unique factorization of ideals, as explained in section 4.3, we can then conclude that the factors
have the �
??????ℎ
power of a principle ideal number since we have �
??????
which is a principle to the �
??????ℎ
power.
To explain this using mathematical notations, let (�+??????
??????�)=??????
??????
be the principle ideal number for ideal
α in the ring ℤ(??????
??????). Therefore, ??????
??????
lies in the ideal class group in the field ℚ(??????
??????) (Mack – Crane, 2015).
However, if the class number of ℚ(??????
??????) is not divisible by � then we can use knowledge from group
theory to argue ?????? itself is the principle element to the ring ℤ(??????
??????). Following this, we must continue using
the original ring, however after the addition of principle ideal numbers the unique factorization holds,
therefore it will hold for the cyclotomic integers that created the ideal numbers. From this, we can
define class numbers. Class numbers are used to measure how far a ring is from unique factorization,
since elements from cyclotomic rings can produce principle ideal numbers, class numbers tells us the
number of ideals added into order to restore the unique factorization (Edwards, 1977, Page 163).
4.4 The proof for Regular Primes
In this final chapter that concludes the proof for regular primes, we will explore the mathematics
involved in solving this case, but we will not go into the full details of the proof. This chapter will give an
idea on how to solve Fermat’s Last Theorem for�=���������??????��(�).
Where Lamé failed to create a general proof, he did however succeed in establishing Fermat’s Last
Theorem. Following from the concept of class numbers, there are still a few odd primes that have class
numbers. These numbers, in the field ℚ(??????
??????) are 3,5,7,11,13,17,19. To salvage this and solve the final
part of the proof we turn to regular primes. We call any prime, � a regular prime if and only if it does not
divide the class number in the field ℚ(??????
??????). By solving for regular primes, we are able to complete the
proof for the theorem (Mack – Crane, 2015).
Theorem 6. The equation �
??????
+�
??????
=�
??????
has no integer solution �,�,� if � is any regular prime.
In order to show the overview of the proof, we will split the theorem into two cases, as shown for the
case of �=3 in chapter three. Whereby both cases show if any of the integers �,�,� are divisible by �.
Case 1: not divisible by ??????
Lemma 6. The integers �,�,� are co-prime and not divisible by �
Proof overview. We can write the equation in Theorem 6. as (Edwards, 1977, Page 168 – 173):
�
??????
=�
??????
+�
??????
=(�+�)(�+??????
??????�)(�+??????
??????
2
�)…(�+??????
??????
??????−1
�).
The first part of this proof is to show that all factors from the above equation are all co-prime. Due to
the unique factorization of ideals, as explained in section 4.3, we can then conclude that the factors
have the �
??????ℎ
power of a principle ideal number since we have �
??????
which is a principle to the �
??????ℎ
power.
To explain this using mathematical notations, let (�+??????
??????�)=??????
??????
be the principle ideal number for ideal
α in the ring ℤ(??????
??????). Therefore, ??????
??????
lies in the ideal class group in the field ℚ(??????
??????) (Mack – Crane, 2015).
However, if the class number of ℚ(??????
??????) is not divisible by � then we can use knowledge from group
theory to argue ?????? itself is the principle element to the ring ℤ(??????
??????). Following this, we must continue using
20
this fact to see if the factorization of the ring causes an infinite descent, thereby concluding that there
are no integer solutions to Fermat’s Last Theorem, where �a regular prime; if the integers �,�,� is not
divisible by � (Edwards, 1977, Page 170).
Case 2: divisible by ??????
Lemma 7. The integers �,�,� are co-prime and divisible by �
Proof overview. Referring to the same equation as in Case 1:
�
??????
=�
??????
+�
??????
=(�+�)(�+??????
??????�)(�+??????
??????
2
�)…(�+??????
??????
??????−1
�).
If � does divide either �,�,�, then this shows that all factors shown are not relatively prime meaning
that all factors are divisible by �. Despite this, the same case follows since the �
??????ℎ
power is principle (�
??????
)
then the factors itself are principle �
??????ℎ
powers. We then factorize elements of ℤ(??????
??????) and form an
infinite descent. Therefore we prove Fermat’s Last Theorem for regular primes with the conclusion that
in both cases, there are no integer solution �,�,� (Mack – Crane, 2015).
this fact to see if the factorization of the ring causes an infinite descent, thereby concluding that there
are no integer solutions to Fermat’s Last Theorem, where �a regular prime; if the integers �,�,� is not
divisible by � (Edwards, 1977, Page 170).
Case 2: divisible by ??????
Lemma 7. The integers �,�,� are co-prime and divisible by �
Proof overview. Referring to the same equation as in Case 1:
�
??????
=�
??????
+�
??????
=(�+�)(�+??????
??????�)(�+??????
??????
2
�)…(�+??????
??????
??????−1
�).
If � does divide either �,�,�, then this shows that all factors shown are not relatively prime meaning
that all factors are divisible by �. Despite this, the same case follows since the �
??????ℎ
power is principle (�
??????
)
then the factors itself are principle �
??????ℎ
powers. We then factorize elements of ℤ(??????
??????) and form an
infinite descent. Therefore we prove Fermat’s Last Theorem for regular primes with the conclusion that
in both cases, there are no integer solution �,�,� (Mack – Crane, 2015).
21
5 Conclusion
The general concept behind Fermat’s Last Theorem is uncomplicated, since it has been regarded as a
topic that a mere 12 year old child can understand (Varma, 2008). Andrew Wiles stated, “Here was a
problem, that I, a ten – year old, could understand and I knew from that moment that I would never let
it go. I had to solve it” (Neil Pieprzak, 2008). However, the interesting and more useful part about the
theorem is not the actual proof, but rather the mathematics used to create it. The theorem caused
substantial advancements in number theory, due to the sheer fact that the ‘simple’ equation introduced
many new mathematical ideas as well as a new branch of mathematics in order to create the proof.
Many of the concepts used, such as ideal principal numbers, showed a creative level of knowledge
needed to solve the theorem. Expanding on this point, we are led to assume many factors in each proof,
such as assuming that the integers are co-prime which shows that a lot of assumptions are created
before tackling the problem. The proofs demonstrated are by far not the most perfect. However, by
comparing all three proofs shown, one can understand the complexity to generalize Fermat’s Last
Theorem. Although all proofs follow a similar trend in that an infinite descent must be acquired, the
calculations of each are different and require some sense of imagination. What we have seen in this
project are the proofs for individual exponents to the theorem. However, Andrew Wiles’s proof used
new mathematical techniques. He used elliptical curves and modular forms rather than opting for the
more traditional and well known use of the infinite descent. The world now knows the general proof to
Fermat’s Last Theorem, but one day there may be a chance that Fermat’s general proof will be
discovered and for that I encourage the reader to pursue this knowledge.
5 Conclusion
The general concept behind Fermat’s Last Theorem is uncomplicated, since it has been regarded as a
topic that a mere 12 year old child can understand (Varma, 2008). Andrew Wiles stated, “Here was a
problem, that I, a ten – year old, could understand and I knew from that moment that I would never let
it go. I had to solve it” (Neil Pieprzak, 2008). However, the interesting and more useful part about the
theorem is not the actual proof, but rather the mathematics used to create it. The theorem caused
substantial advancements in number theory, due to the sheer fact that the ‘simple’ equation introduced
many new mathematical ideas as well as a new branch of mathematics in order to create the proof.
Many of the concepts used, such as ideal principal numbers, showed a creative level of knowledge
needed to solve the theorem. Expanding on this point, we are led to assume many factors in each proof,
such as assuming that the integers are co-prime which shows that a lot of assumptions are created
before tackling the problem. The proofs demonstrated are by far not the most perfect. However, by
comparing all three proofs shown, one can understand the complexity to generalize Fermat’s Last
Theorem. Although all proofs follow a similar trend in that an infinite descent must be acquired, the
calculations of each are different and require some sense of imagination. What we have seen in this
project are the proofs for individual exponents to the theorem. However, Andrew Wiles’s proof used
new mathematical techniques. He used elliptical curves and modular forms rather than opting for the
more traditional and well known use of the infinite descent. The world now knows the general proof to
Fermat’s Last Theorem, but one day there may be a chance that Fermat’s general proof will be
discovered and for that I encourage the reader to pursue this knowledge.
22
6 Evaluation
This dissertation has tested my mathematical knowledge and skills as well as my sanity, in the sense that
all effort was put into this project. However, as much as this topic has been very challenging for me, it
was one of the main reasons why I decided to study this course, especially after reading Simon Singh’s
book ‘Fermat’s Last Theorem (2002)’ whilst studying for my GCSEs. Having said that this project tested
my sanity, I did enjoy the research and understanding how the mind of all the mathematicians
essentially worked together to find the proof to the theorem, whilst learning the actual mathematics
involved to the best of my ability.
The research aspects for my project was challenging due to the difficult mathematics involved with the
‘simple’ looking equation. To put it into perspective, it did take almost 4 centuries to solve, whereby
Andrew Wiles only managed to solve it 31 years after learning about the conjecture. Many of the books
and online resources, such as blogs and PDFs that I found provided the proofs I needed, but with the
drawback that each proof for each exponent of � was slightly different when comparing resources; such
as the proof for �=4, when comparing the books ‘Fermat’s Last Theorem for Amateurs’ by Paulo
Ribenboim to ‘Fermat’s Last Theorem: A Genetic Introduction to Algebraic Number Theory’ by Harold M.
Edwards, had some minor differences. Therefore in order to understand each proof I had to constantly
read the same line or paragraph over and over.
Expanding on this issue, many of the proofs from the resources I used had different notations to the
same proof. An example of this is the use of �
�
=1 for roots of unity in Harold M. Edwards’s book,
rather than ??????
�
=1 which is used by Ila Varma in her paper and Ernst Kummer himself, whom
introduced the complex number. Even though this example is not as extreme, I decided to stick to the
resources which gave me more understanding to the problem at hand.
After understanding each theorem and proof, the next difficulty was making sure that each chapter I
wrote flowed rather than having a completely different topic. To achieve this I had to write many
subchapters, with each explaining the method I would utilize before explaining the proof. An example of
this was when I explained the biquadratic equation and method of infinite descent before giving the
proof of �=4, whilst introducing the Euclid’s Lemma in-between the proof.
My goal for this project was to ultimately create the proof that Fermat had claimed to have had. From
my research I learnt that the mathematics used by Andrew Wiles, in his general proof, was not available
in Fermat’s time and that it was only achievable due to the ever-growing knowledge and input by
numerous mathematicians. I wanted to see if I could create this lost proof, but as my supervisor pointed
6 Evaluation
This dissertation has tested my mathematical knowledge and skills as well as my sanity, in the sense that
all effort was put into this project. However, as much as this topic has been very challenging for me, it
was one of the main reasons why I decided to study this course, especially after reading Simon Singh’s
book ‘Fermat’s Last Theorem (2002)’ whilst studying for my GCSEs. Having said that this project tested
my sanity, I did enjoy the research and understanding how the mind of all the mathematicians
essentially worked together to find the proof to the theorem, whilst learning the actual mathematics
involved to the best of my ability.
The research aspects for my project was challenging due to the difficult mathematics involved with the
‘simple’ looking equation. To put it into perspective, it did take almost 4 centuries to solve, whereby
Andrew Wiles only managed to solve it 31 years after learning about the conjecture. Many of the books
and online resources, such as blogs and PDFs that I found provided the proofs I needed, but with the
drawback that each proof for each exponent of � was slightly different when comparing resources; such
as the proof for �=4, when comparing the books ‘Fermat’s Last Theorem for Amateurs’ by Paulo
Ribenboim to ‘Fermat’s Last Theorem: A Genetic Introduction to Algebraic Number Theory’ by Harold M.
Edwards, had some minor differences. Therefore in order to understand each proof I had to constantly
read the same line or paragraph over and over.
Expanding on this issue, many of the proofs from the resources I used had different notations to the
same proof. An example of this is the use of �
�
=1 for roots of unity in Harold M. Edwards’s book,
rather than ??????
�
=1 which is used by Ila Varma in her paper and Ernst Kummer himself, whom
introduced the complex number. Even though this example is not as extreme, I decided to stick to the
resources which gave me more understanding to the problem at hand.
After understanding each theorem and proof, the next difficulty was making sure that each chapter I
wrote flowed rather than having a completely different topic. To achieve this I had to write many
subchapters, with each explaining the method I would utilize before explaining the proof. An example of
this was when I explained the biquadratic equation and method of infinite descent before giving the
proof of �=4, whilst introducing the Euclid’s Lemma in-between the proof.
My goal for this project was to ultimately create the proof that Fermat had claimed to have had. From
my research I learnt that the mathematics used by Andrew Wiles, in his general proof, was not available
in Fermat’s time and that it was only achievable due to the ever-growing knowledge and input by
numerous mathematicians. I wanted to see if I could create this lost proof, but as my supervisor pointed
23
out we are uncertain if Fermat ever created this proof. Due to my lack of knowledge in number theory
as well as my mathematical capability, this was much further from achievable than I first anticipated.
Therefore I decided to leave out my creation of the general proof and focus on introducing the theorem
and exploring the mathematics involved in the finding the proof for important special cases such as �=
4,3 and regular primes. This again, was proven to be difficult, especially with regular primes since I had
to understand other mathematical theorems and proofs, such as cyclotomic integers and class numbers.
In terms of writing the content of this dissertation, my biggest problem, was writing it in order for level 6
students to understand. Initially I had written the proofs based on what I learnt and understood from
the resources I had used, but I had only realised that these resources assume that the reader would
have background knowledge after my supervisor told me to explain what terms such as co-prime meant.
This in turn would mean that I had to expand on the proofs and fill in the gaps in knowledge that was
left out by the resources I used, meaning that the proofs I have shown in this dissertation are longer
than those in online resources and books.
My overall aim of the project was to introduce, explore and explain the mathematics involved from the
creation of Fermat’s Last Theorem through to, but not including, Andrew Wiles’s general proof and
explain the theorem, with the added aim to make it more accessible to students that do not have prior
knowledge of number theory. However, this aim changed due to my initial lack of research. I did not
realise that the theorem was more in-depth and much harder until it was almost too late. Having wasted
a lot of time in reading books that provided more information on the historical events rather than the
mathematical events, I had to quickly change my goals. For example, I have been pushing myself to
write the full detailed proof for regular primes but this became an unrealistic goal since I was not able to
understand the theorem enough to write a lot about it. The detailed proof was arduous for me to
understand, since I do not have any knowledge on number theory, therefore in writing this project, I
have taught myself the basics of number theory in terms of Fermat’s Last Theorem.
I believe that I have achieved my aims and provided that correct information to allow level 6 students or
any other reader to understand the theorem, and how it aided the advancements of number theory.
Knowing that I was going to write my dissertation on Fermat’s Last Theorem ever since starting
university, I wish I had prepared myself by doing the necessary research that would have allowed me to
complete this project with ease, and given me more chances to generate my own proof. Nevertheless, I
have enjoyed reading and learning about Fermat’s Last Theorem and the evolution of number theory.
out we are uncertain if Fermat ever created this proof. Due to my lack of knowledge in number theory
as well as my mathematical capability, this was much further from achievable than I first anticipated.
Therefore I decided to leave out my creation of the general proof and focus on introducing the theorem
and exploring the mathematics involved in the finding the proof for important special cases such as �=
4,3 and regular primes. This again, was proven to be difficult, especially with regular primes since I had
to understand other mathematical theorems and proofs, such as cyclotomic integers and class numbers.
In terms of writing the content of this dissertation, my biggest problem, was writing it in order for level 6
students to understand. Initially I had written the proofs based on what I learnt and understood from
the resources I had used, but I had only realised that these resources assume that the reader would
have background knowledge after my supervisor told me to explain what terms such as co-prime meant.
This in turn would mean that I had to expand on the proofs and fill in the gaps in knowledge that was
left out by the resources I used, meaning that the proofs I have shown in this dissertation are longer
than those in online resources and books.
My overall aim of the project was to introduce, explore and explain the mathematics involved from the
creation of Fermat’s Last Theorem through to, but not including, Andrew Wiles’s general proof and
explain the theorem, with the added aim to make it more accessible to students that do not have prior
knowledge of number theory. However, this aim changed due to my initial lack of research. I did not
realise that the theorem was more in-depth and much harder until it was almost too late. Having wasted
a lot of time in reading books that provided more information on the historical events rather than the
mathematical events, I had to quickly change my goals. For example, I have been pushing myself to
write the full detailed proof for regular primes but this became an unrealistic goal since I was not able to
understand the theorem enough to write a lot about it. The detailed proof was arduous for me to
understand, since I do not have any knowledge on number theory, therefore in writing this project, I
have taught myself the basics of number theory in terms of Fermat’s Last Theorem.
I believe that I have achieved my aims and provided that correct information to allow level 6 students or
any other reader to understand the theorem, and how it aided the advancements of number theory.
Knowing that I was going to write my dissertation on Fermat’s Last Theorem ever since starting
university, I wish I had prepared myself by doing the necessary research that would have allowed me to
complete this project with ease, and given me more chances to generate my own proof. Nevertheless, I
have enjoyed reading and learning about Fermat’s Last Theorem and the evolution of number theory.
24
7 Bibliography
Eberhart, C. (1999) Fermat and His Method of Infinite Descent [Online], Available:
<http://www.ms.uky.edu/~carl/ma330/html/bailey21.html> [Accessed 23rd November 2015]
Edwards, H. M. (1977) Fermat’s last theorem: A genetic introduction to algebraic number theory,
Germany: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Freeman, L. (2005) Fermat’s last theorem [Online], Available:
<http://fermatslasttheorem.blogspot.co.uk/> [Accessed 10th November 2015]
Mack-Crane, S. (2015) Fermat’s last theorem for regular primes, [Online] . Available:
<https://math.berkeley.edu/~sander/speaking/22September2015%20WIM%20Talk.pdf> [Accessed 4th
May 2016]
O’Connor, J. J. and Robertson, E. F. (1999) Euclid biography [Online], Available: <http://www-
groups.dcs.st-and.ac.uk/~history/Biographies/Euclid.html> [Accessed 3rd May 2016]
Pieprzak, N. (2008) Fermat’s last theorem and Andrew wiles [Online], Available:
<https://plus.maths.org/content/fermats-last-theorem-and-andrew-wiles> [Accessed 18th April 2016]
Ribenboim, P. (2013) Fermat’s last theorem for amateurs, United States: Springer-Verlag New York
Singh, S. (2002) Fermat’s last theorem: The story of a riddle that confounded the world's greatest minds
for 358 years, London: Fourth Estate
Singh, S. and Ribet, K. A. (1997) Fermat’s Last Stand, [Online] . Available:
<http://www.fis.cinvestav.mx/~lmontano/sciam/Fermat-SC1197-68.pdf> [Accessed 25th April 2016]
Varma, I. (2008) KUMMER, REGULAR PRIMES, AND FERMAT’S LAST THEOREM, [Online] . Available:
<http://www.math.harvard.edu/~ila/Kummer.pdf> [Accessed 4th April 2016]
7 Bibliography
Eberhart, C. (1999) Fermat and His Method of Infinite Descent [Online], Available:
<http://www.ms.uky.edu/~carl/ma330/html/bailey21.html> [Accessed 23rd November 2015]
Edwards, H. M. (1977) Fermat’s last theorem: A genetic introduction to algebraic number theory,
Germany: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Freeman, L. (2005) Fermat’s last theorem [Online], Available:
<http://fermatslasttheorem.blogspot.co.uk/> [Accessed 10th November 2015]
Mack-Crane, S. (2015) Fermat’s last theorem for regular primes, [Online] . Available:
<https://math.berkeley.edu/~sander/speaking/22September2015%20WIM%20Talk.pdf> [Accessed 4th
May 2016]
O’Connor, J. J. and Robertson, E. F. (1999) Euclid biography [Online], Available: <http://www-
groups.dcs.st-and.ac.uk/~history/Biographies/Euclid.html> [Accessed 3rd May 2016]
Pieprzak, N. (2008) Fermat’s last theorem and Andrew wiles [Online], Available:
<https://plus.maths.org/content/fermats-last-theorem-and-andrew-wiles> [Accessed 18th April 2016]
Ribenboim, P. (2013) Fermat’s last theorem for amateurs, United States: Springer-Verlag New York
Singh, S. (2002) Fermat’s last theorem: The story of a riddle that confounded the world's greatest minds
for 358 years, London: Fourth Estate
Singh, S. and Ribet, K. A. (1997) Fermat’s Last Stand, [Online] . Available:
<http://www.fis.cinvestav.mx/~lmontano/sciam/Fermat-SC1197-68.pdf> [Accessed 25th April 2016]
Varma, I. (2008) KUMMER, REGULAR PRIMES, AND FERMAT’S LAST THEOREM, [Online] . Available:
<http://www.math.harvard.edu/~ila/Kummer.pdf> [Accessed 4th April 2016]
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