perfect numbers and mersenne primes
TRANSCRIPT
Perfect Numbers
Abundant Numbers
Deficient Numbers
Perfect Number: The proper divisors of a number are all its divisors excluding the number itself.
Mersenne Primes
121, 2, 3, 4, 6, 12
1 + 2 + 3 + 4 + 6 = 16
16 > 12
181, 2, 3, 6, 9, 18
1 + 2 + 3 + 6 + 9 = 21
21 > 18
151, 3, 5, 15
1 + 3 + 5 = 9
9 < 15
Abundant Number
Abundant Number
Deficient Number
12 18 15Abundant Abundant Deficient
61, 2, 3, 6
1 + 2 + 3 = 6
6 = 6 Perfect Number
Perfect Number
P1 = 6
The Mathematicians of Ancient Greece.
Pythagoras (570 – 500 BC.)
Euclid of (325 – 265 BC.)
Archimedes (287 – 212 BC.)
Eratosthenes (275-192 BC.)
P1 = 6
P2 = 28
P3 = 496
P4 = 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 P5
The Mathematicians of Ancient Greece.
Pythagoras (570 – 500 BC.)
Euclid of (325 – 265 BC.)
Archimedes (287 – 212 BC.)
Eratosthenes (275-192 BC.)
P1 = 6
P2 = 28
P3 = 496
P4 = 8128
P5 =? (a 5 digit number?)
P5 = 33 550 336 (8 digits)
P5 = 33 550 336 (1456 Not Known) 8 digits
P6 = 8 589 869 056 (1588 Cataldi) 10 digits
P7 = 137 438 691 328 (1588 Cataldi) 12 digits
P8 = 2 305 843 008 139 952 128 (1772 Euler) 19 digits
P9 = 2 658 455 991 569 831 744 654 692 615 953 842 176 (1883 Pervushin) 37 digits
P10 = 191 561 942 608 236 107 294 793 378 084 303 638 130 997 321 548 169 216 (1911: Powers) 54 digits
P11 =13 164 036 458 569 648 337 239 753 460 458 722 910 223 472 318386 943 117 783 728 128 (1914 Powers) 65 digits
P12 =14 474 011 154 664 524 427 946 373 126 085 988 481 573 677 491474 835 889 066 354 349 131 199 152 128 (1876 Edouard Lucas) 77 digits
P13=23562723457267347065789548996709904988477547858392600710143020528925780432155433824984287771524270103944969186640286445341759750633728317862223973036553960260056136025556646250327017528033831439790236838624033171435922356643219703101720713163527487298747400647801939587165936401087419375649057918549492160555646 976 (1952 Robinson) 314 digits
P1= 6 P2= 28 P3= 496 P4 = 8128
Mersenne Primes
A Mersenne number is any number of the form 2n – 1
21 – 1 = 1 22 – 1 =3 23 – 1 =7 24 – 1 =15 25 – 1 =31
26 – 1 = 63 27 – 1 = 127 28 – 1 = 255 29 – 1 = 511
210 – 1 = 1023 211 – 1 = 2047 212 – 1 = 4095
Mersenne Primes
A Mersenne number is any number of the form 2n – 1
21 – 1 = 1 22 – 1 = 3 23 – 1 = 7 24 – 1 = 15 25 – 1 = 31
26 – 1 = 63 27 – 1 = 127 28 – 1 = 255 29 – 1 = 511
210 – 1 = 1023 211 – 1 = 2047 212 – 1 = 4095
Mersenne Primes
A French monk called Marin Mersenne stated in one of his books in 1644 that for the primes:
2n – 1
n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257
22 – 1 = 3 23 – 1 = 7 25 – 1 = 31 27 – 1 = 127
1588 - 1644
Mersenne Primes
22 – 1 = 3 23 – 1 = 7 25 – 1 = 31 27 – 1 = 127
1588 - 1644
213 – 1 217 – 1
219 – 1 231 – 1 2127 – 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
261 – 1 289 – 1 2107 – 1
In subsequent years various mathematicians showed that his conjecture was not correct.
Mersenne Primes and Perfect Numbers
22 – 1 = 3 23 – 1 = 7 25 – 1 = 31 27 – 1 = 127
1588 - 1644
213 – 1 217 – 1
219 – 1 231 – 1 2127 – 1 261 – 1 289 – 1 2107 – 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 2n -1 x 2n-1 Perfect Number
2 3 x ? 6
3 7 x ? 28
5 31 x ? 496
7 127 x ? 8128
Mersenne Primes and Perfect Numbers
22 – 1 = 3 23 – 1 = 7 25 – 1 = 31 27 – 1 = 127
1588 - 1644
213 – 1 217 – 1
219 – 1 231 – 1 2127 – 1 261 – 1 289 – 1 2107 – 1
n 2n -1 x 2n-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.
Mersenne Primes and Perfect Numbers
22 – 1 = 3 23 – 1 = 7 25 – 1 = 31 27 – 1 = 127
1588 - 1644
213 – 1 217 – 1
219 – 1 231 – 1 2127 – 1 261 – 1 289 – 1 2107 – 1
n 2n -1 x 2n-1 Perfect Number
2 3 x 2 6 21
3 7 x 4 28 22
5 31 x 16 496 24
7 127 x 64 8128 26
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.
Mersenne Primes and Perfect Numbers
22 – 1 = 3 23 – 1 = 7 25 – 1 = 31 27 – 1 = 127
1588 - 1644
213 – 1 217 – 1
219 – 1 231 – 1 2127 – 1 261 – 1 289 – 1 2107 – 1
If 2n -1 is a Mersenne prime then 2n – 1 x 2n-1 is a perfect number. Check this for the first few.
22 – 1 x 21 = 3 x 2 = 6
23 – 1 x 22 = 7 x 4 = 28
25 – 1 x 24 = 31 x 16 = 496
27 – 1 x 26 = 127 x 64 = 8128
Mersenne Primes and Perfect Numbers
22 – 1 = 3 23 – 1 = 7 25 – 1 = 31 27 – 1 = 127
1588 - 1644
213 – 1 217 – 1
219 – 1 231 – 1 2127 – 1 261 – 1 289 – 1 2107 – 1
Mersenne Primes and Perfect Numbers
22 – 1 = 3 23 – 1 = 7 25 – 1 = 31 27 – 1 = 127
1588 - 1644
213 – 1 217 – 1
219 – 1 231 – 1 2127 – 1 261 – 1 289 – 1 2107 – 1
Research other information about Mersenne Primes and Perfect Numbers
http://www.mersenne.org/
