## Contents

### Mersenne prime number

1. 2: 2^2-1 = 3
2. 3: 2^3-1 = 7
3. 5: 2^5-1 = 31
4. 7: 2^7-1 = 127
5. 11: 2^11-1 = 2047 = 23 * 89 (composite number)
6. 13: 2^13-1 = 8191
7. 17: 2^17-1 = 131071
8. 19: 2^19-1 = 524287
9. ...
10. 67: 2^67-1 = 147573952589676412927 = 193707721 * 761838257287 (composite number)
11. ...

### Amicable Numbers(Japanese:Shinwasuu, Yuusuu(Yuuaisuu))

The sum of factors is each other number.

1. 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
2. 1184, 1210
3. 17296, 18416
4. 9437056, 9363584
5. ...

### Complete numbers (Perfect numbers)

The sum of factors is the original number.

1. 6 : { 1 2 3 }, 1 + 2 + 3 = 6
2. 28 : { 1 2 4 7 14 }, 1 + 2 + 4 + 7 + 14 = 28
3. 496
4. 8128
5. 33550336
6. ...

### Ramanujan 1

• 1729 = 12^3 + 1^3 = 10^3 + 9^3

### Euler 1

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?

### Fibonacci 1

• 3794765361567513, 20615674205555510

### The biography of Diophantus of Alexandria.

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.)