Pierre de Fermat

An overview of Pierre de Fermat's life.

1. Fermatting with
Numbers
By: Professor April Harning
History of Mathematics Course
Chapter 10: 17
th
Century

2. That’s right!
Pierre de Fermat (1601-1665)

3. Fermat was a…
•Family man
•Lawyer
•Government official
•Mathematician

4. Fermat’s Life
•Born in France
•Dominique
•Claire de Long
•Siblings
•Married Louise de
Long

5. Fermat’s Studies
•Classical literature
•Languages
•Law
•Attended…

6. Became a …
•Purchased part of Toulouse’s
parliament
•Counsellor of Local Parliament
•Criminal Court
•Chief Spokesman of the Grand
Chamber

7. Time for Mathematics
•Retired
•Contributions
–Geometry
–Infinitesimal calculus
–Theory of numbers
–Probability
•His friends

8. Fermat’s Geometry
•Old Works
•Plane Loci
•Analytic Geometry
•Rotation of Axes
•Cubic and Quartic
Equations
•Higher Dimension Curves
•De Linearum Curvarum
cum Lineis Rectis

9. Fermat’s Calculus
•Hyperbolas and Parabolas of Fermat
•Archimedean Spiral, r = aΘ
•Differentiation
•Maximum and Minimum Values
•“Fermat, the true inventor of the differential
calculus.” ~Laplace
•Integral Calculus
.0
)()(
lim
0
=
-+
®
E
xfExf
E

10. Fermat’s Number Theory
•Arithmetica
•Method of Infinite Descent
•Girard’s Assertion
•Fermat’s Little Theorem
•Fermat Numbers
65537F
4
257F
3
17F
2
5F
1
3F
0
PrimesFermat
12
2
+=
n
n
F

11. Proof of Fermat’s Little Theorem
Theorem: Let p be a prime and suppose
that p does not divide a. Then,
Given: Prime p where p does not divide a
Prove:
•Consider the first p-1 positive multiples
of a, (a, 2a, 3a, …,(p-1)a)
•
• IMPOSSIBLE
).(mod1
1
pa
p
º
-
11),(mod -£<£º psrpsara
)(modpsrº
).(mod1
1
pa
p
º
-

12. Proof of Fermat’s Little Theorem
•Previous set of integers must be
congruent modulo p to 1, 2, 3, …, p-1
•
•
•Therefore, because p
does not divide (p-1)!
)(mod1
1
pa
p
º
-
)(mod)!1()!1(
1
pppa
p
-º-
-
))(mod1(321)1(32 ppapaaa -×××××=-×××××

13. Fermat’s Probability
•Gambler’s Dispute
with Pascal
•Chevalier de Méré
informed Pascal
•Exchange of Letters

14. Fermat’s Last Theorem
•x
n
+ y
n
= z
n
has no integer solutions for x, y,
and z when n > 2
•Famous quote, "I have discovered a truly
remarkable proof which this margin is too
small to contain.“
•Challenged mathematicians
•In 1994, Andrew Wiles and Richard Taylor

15. Peaceful Death
•In 1665, …
•City of Castres
•Samuel, his son
•Accuracy of work
•Publications