# Proof of infinite prime numbers

#### — A mathematical proof that prime numbers occur indefinitely along the number sequence —

## Prime numbers; a visual aid

In order to find the prime numbers from a certain number (9) to twice that same number (2*9 = 18) we only need the prime numbers from 2 to 9. Map those primes on the vertical and the number sequence on the horizontal. Mark every multiple of the listed primes (for 2 that's 2 and 2+2 and 2+2+2 etc.). When all prime multiples are charted look for marks in every vertical column from 10 to 18. Those columns without any marks are the prime numbers from 10 to 18.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | |

2 | ||||||||||||||||||

3 | ||||||||||||||||||

5 | ||||||||||||||||||

7 | ||||||||||||||||||

→ | → |

## Calculating the amount of primes between N/2 and N

- Let a0 be 1.
- Let a1 be the value of the first prime number (=2).
- Let a2 be the value of the second prime number (=3).
- Let ap be the value of prime number p.
- Let Ai be the amount of numbers identified to be not primes by the isofactor of i's.

A1 = |N/a0*a1|

A2 = |N/a0*a2| - |N/a1*a2|

A3 = |N/a0*a3| - |N/a1*a3| - |N/a2*a3| + |N/a1*a2*a3|

Etcetera, up to Ap.

The amount of primes between N/2 and N equals N-1-A.

If this amount is to be zero (that is after the hypothetical final prime number) then 0 = N-1-A. In this case A = N-1. When the prime numbers become very large, N becomes very large but the a*a-phrase becomes larger faster. The number A will never be large enough to be N-1.