# Euler's proof of infinite primes first since Euclid?

Q. Is it true that Euler's proof of infinite primes was the first since Euclid's which was from around 300BC?

Note: By Euler's proof, I mean the use of his Euler product formula for the zeta function $$\zeta(s)$$ which diverges as $$s\rightarrow\infty$$, implying an infinitude of factors in the product, which means an infinite of primes.

• what you meant in your question is that the zeta function $\zeta (s)$ diverges when $s=1$ (in other words, the harmonic series diverges as $n$ tends to $\infty$) and not when $s$ tends to infinity (it actually converges for $s>1$). Mar 14, 2021 at 10:26

## 1 Answer

There is a proof by Goldbach, which appears in a letter he wrote to Euler in 1730 (a few years before Euler published his product formula for the zeta function). It is as follows: if $$F_n=2^{2^n}+1$$ (that is, if $$F_n$$ is the $$n$$th Fermat number), then$$F_n=F_1F_2\ldots F_{n-1}+2.\tag1$$Since each $$F_n$$ is odd, it follows from $$(1)$$ that, if $$n\ne m$$, then $$F_n$$ and $$F_m$$ are relatively prime. So, take a prime factor $$p_n$$ of $$F_n$$, and the set $$\{p_1,p_2,p_3,\ldots\}$$ will be an infinite set of prime numbers.

• Thanks @josé-carlos-santos - that is very helpful. Do you have a reference for when Euler published his product formula and in which book? I'm keen to reproduce the original texts as part of teaching. Mar 14, 2021 at 23:03
• also thanks to Jose's comment I found this primes.utm.edu/notes/proofs/infinite/goldbach.html Mar 14, 2021 at 23:06
• You will find the complete reference here. Mar 14, 2021 at 23:07
• A thread on MO regarding Goldbach's proof of the infinitude of primes: mathoverflow.net/questions/22316/… Mar 22, 2021 at 1:10