Perfect numbers and mersenne primes

Perfect numbers and mersenne primes

1. University of Zakho Dept : Mathematics Name : Pasar H.Ibrahim Subject : Perfect Number

2. Perfect Numbers Abundant Numbers Deficient Numbers Perfect Number: The proper divisors of a number are all its divisors excluding the number itself. Mersenne Primes

3. 12 1, 2, 3, 4, 6, 12 1 + 2 + 3 + 4 + 6 = 16 16 > 12 18 1, 2, 3, 6, 9, 18 1 + 2 + 3 + 6 + 9 = 21 21 > 18 15 1, 3, 5, 15 1 + 3 + 5 = 9 9 < 15 Abundant Number Abundant Number Deficient Number

4. 12 18 15 Abundant Abundant Deficient 6 1, 2, 3, 6 1 + 2 + 3 = 6 6 = 6  Perfect Number Perfect Number P 1 = 6

5. The Mathematicians of Ancient Greece. Pythagoras ( 570 – 500 BC.) Euclid of ( 325 – 265 BC.) Archimedes ( 287 – 212 BC.) Eratosthenes ( 275-192 BC.) P 1 = 6 P 2 = 28 P 3 = 496 P 4 = 8128 1 + 2 + 3 = 6 1 + 2 + 4 + 7 + 14 = 28 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128 The mathematicians of Ancient Greece knew the first 4 perfect numbers and the search was on for the P 5

6. The Mathematicians of Ancient Greece. Pythagoras ( 570 – 500 BC.) Euclid of ( 325 – 265 BC.) Archimedes ( 287 – 212 BC.) Eratosthenes ( 275-192 BC.) P 1 = 6 P 2 = 28 P 3 = 496 P 4 = 8128 P 5 =? (a 5 digit number?) P 5 = 33 550 336 (8 digits)

7. P 5 = 33 550 336 (1456 Not Known) 8 digits P 6 = 8 589 869 056 (1588 Cataldi ) 10 digits P 7 = 137 438 691 328 (1588 Cataldi ) 12 digits P 8 = 2 305 843 008 139 952 128 (1772 Euler) 19 digits P 9 = 2 658 455 991 569 831 744 654 692 615 953 842 176 (1883 Pervushin ) 37 digits P 10 = 191 561 942 608 236 107 294 793 378 084 303 638 130 997 321 548 169 216 (1911: Powers) 54 digits P 11 =13 164 036 458 569 648 337 239 753 460 458 722 910 223 472 318 386 943 117 783 728 128 (1914 Powers) 65 digits P 12 =14 474 011 154 664 524 427 946 373 126 085 988 481 573 677 491 474 835 889 066 354 349 131 199 152 128 (1876 Edouard Lucas) 77 digits P 13 =23562723457267347065789548996709904988477547858392600710143020528925780432155433824984287771524270103944969186640286445341759750633728317862223973036553960260056136025556646250327017528033831439790236838624033171435922356643219703101720713163527487298747400647801939587165936401087419375649057918549492160555646 976 (1952 Robinson) 314 digits P 1 = 6 P 2 = 28 P 3 = 496 P 4 = 8128

8. Mersenne Primes A Mersenne number is any number of the form 2 n – 1 2 1 – 1 = 1 2 2 – 1 =3 2 3 – 1 =7 2 4 – 1 =15 2 5 – 1 =31 2 6 – 1 = 63 2 7 – 1 = 127 2 8 – 1 = 255 2 9 – 1 = 511 2 10 – 1 = 1023 2 11 – 1 = 2047 2 12 – 1 = 4095

9. Mersenne Primes A Mersenne number is any number of the form 2 n – 1 2 1 – 1 = 1 2 2 – 1 = 3 2 3 – 1 = 7 2 4 – 1 = 15 2 5 – 1 = 31 2 6 – 1 = 63 2 7 – 1 = 127 2 8 – 1 = 255 2 9 – 1 = 511 2 10 – 1 = 1023 2 11 – 1 = 2047 2 12 – 1 = 4095

10. Mersenne Primes A French monk called Marin Mersenne stated in one of his books in 1644 that for the primes: 2 n – 1 n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644

11. Mersenne Primes 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644 2 13 – 1 2 17 – 1 2 19 – 1 2 31 – 1 2 127 – 1 n = 2, 3, 5, 7 , 13 , 17 , 19, 31 , 61 , 89 , 107 , 127, n = 2, 3, 5, 7 , 13 , 17 , 19, 31 , 67, 127, and 257 Mersenne’s List Completed List 2 61 – 1 2 89 – 1 2 107 – 1 In subsequent years various mathematicians showed that his conjecture was not correct.

12. Mersenne Primes and Perfect Numbers 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644 2 13 – 1 2 17 – 1 2 19 – 1 2 31 – 1 2 127 – 1 2 61 – 1 2 89 – 1 2 107 – 1 There is a formula linking a Mersenne Prime to its corresponding perfect number by multiplication. Use the table below to help you find it. n 2 n -1 x 2 n-1 Perfect Number 2 3 x ? 6 3 7 x ? 28 5 31 x ? 496 7 127 x ? 8128

13. Mersenne Primes and Perfect Numbers 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644 2 13 – 1 2 17 – 1 2 19 – 1 2 31 – 1 2 127 – 1 2 61 – 1 2 89 – 1 2 107 – 1 n 2 n -1 x 2 n-1 Perfect Number 2 3 x 2 6 3 7 x 4 28 5 31 x 16 496 7 127 x 64 8128 Write as a power of 2 There is a formula linking a Mersenne Prime to its corresponding perfect number by multiplication. Use the table below to help you find it.

14. Mersenne Primes and Perfect Numbers 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644 2 13 – 1 2 17 – 1 2 19 – 1 2 31 – 1 2 127 – 1 2 61 – 1 2 89 – 1 2 107 – 1 n 2 n -1 x 2 n-1 Perfect Number 2 3 x 2 6 2 1 3 7 x 4 28 2 2 5 31 x 16 496 2 4 7 127 x 64 8128 2 6 Write as a power of 2 There is a formula linking a Mersenne Prime to its corresponding perfect number by multiplication. Use the table below to help you find it.

15. Mersenne Primes and Perfect Numbers 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644 2 13 – 1 2 17 – 1 2 19 – 1 2 31 – 1 2 127 – 1 2 61 – 1 2 89 – 1 2 107 – 1 If 2 n -1 is a Mersenne prime then 2 n – 1 x 2 n-1 is a perfect number. Check this for the first few. 2 2 – 1 x 2 1 = 3 x 2 = 6 2 3 – 1 x 2 2 = 7 x 4 = 28 2 5 – 1 x 2 4 = 31 x 16 = 496 2 7 – 1 x 2 6 = 127 x 64 = 8128

16. NEWS FLASH 4 th September 2006 44 th Mersenne Prime Found. 2 32 582 657 – 1 has 9,808,358 digits Mersenne Primes and Perfect Numbers 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644 2 13 – 1 2 17 – 1 2 19 – 1 2 31 – 1 2 127 – 1 2 61 – 1 2 89 – 1 2 107 – 1

17. Mersenne Primes and Perfect Numbers 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644 2 13 – 1 2 17 – 1 2 19 – 1 2 31 – 1 2 127 – 1 2 61 – 1 2 89 – 1 2 107 – 1 Research other information about Mersenne Primes and Perfect Numbers http://www.mersenne.org/

18. Thank you For listening By:Pasar H.Ibrahim

19. ? By:Pasar H.Ibrahim