Math

# Infinity of prime numbers.

## Proving the infinity of prime numbers, noted as P.

In this post we will prove the infinity of prime numbers, noted as $P$.

First, what is a prime number? a prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself, e.g. 2, 3, 5, 71, 97 …

Throughout history, multiple mathematicians studied this branch of mathematics, particularly Euclid, Gauss, and Riemann. Euclid proved this mathematical property by multiplying all primes from 1 to n and adding 1.

Hence the property seems obvious. In this post we will try Gauss’s method. In order to do that, we need these notions:

First let’s suppose that, the set of prime numbers is finite, which means that we can enumerate the elements of this set:

$P = {P_1, P_2, ..., P_r}$ with $r = card(P)$

And we consider this series: $S_n = \sum_{i=1}^{k} 1/k$

If we apply the fundamental theorem of arithmetic, we find

Now, we will try to find a max for this series $S_n$. For this purpose let’s simplify our formula

And let’s consider also this value:

with $\alpha_i the max of exponent of P_i$

If we take one of this $\alpha_i$ and call it $\beta$, we then have

We can then conclude that $% $ And since terms in $t$ are geometric sequences. We find then:

and this quantity is finite, However since harmonic series diverge, we find that: $% $

Which is absurd, hence we can say that prime numbers are infinite.