- Paul Erd"os: The man who loved only numbers
- Mersenne prime number
- Amicable numbers
- Complete numbers
- Ramanujan 1
- Euler 1
- Fibonacci 1
- Diophantus

- 2: 2^2-1 = 3
- 3: 2^3-1 = 7
- 5: 2^5-1 = 31
- 7: 2^7-1 = 127
- 11: 2^11-1 = 2047 = 23 * 89 (composite number)
- 13: 2^13-1 = 8191
- 17: 2^17-1 = 131071
- 19: 2^19-1 = 524287
- ...
- 67: 2^67-1 = 147573952589676412927 = 193707721 * 761838257287 (composite number)
- ...

The sum of factors is each other number.

- 220, 284

220 = { 1 2 4 5 10 11 20 22 44 55 110 }

(+ 1 2 4 5 10 11 20 22 44 55 110) -> 284

284 = { 1 2 4 71 142 }

(+ 1 2 4 71 142 ) -> 220

- 1184, 1210
- 17296, 18416
- 9437056, 9363584
- ...

The sum of factors is the original number.

- 6 : { 1 2 3 }, 1 + 2 + 3 = 6
- 28 : { 1 2 4 7 14 }, 1 + 2 + 4 + 7 + 14 = 28
- 496
- 8128
- 33550336
- ...

- 1729 = 12^3 + 1^3 = 10^3 + 9^3

These are prime numbers generating function. (Only in some range)

- n^2 + n + 17, n = [0, 15]
- n^2 + n + 41, n = [0, 39]
- The valid range is the last term -2. It should be -1 is easy to uderstand. Because the same number makes even number. But why -2?

- 3794765361567513, 20615674205555510

His boyhood lasted 1/6th of his life; he married after 1/7th more; his beard grew after 1/12th more, and his son was born 5 years later; the son lived to half his father's age, and the father died 4 years after the son.

(You know this is also a Diophantus equation.)

Copyright (C) 2003 YAMAUCHI Hitoshi