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.

  • $\begingroup$ 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 $s>1$). $\endgroup$
    – user2554
    Mar 14, 2021 at 10:26

1 Answer 1


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.

  • $\begingroup$ 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. $\endgroup$
    – Penelope
    Mar 14, 2021 at 23:03
  • $\begingroup$ also thanks to Jose's comment I found this primes.utm.edu/notes/proofs/infinite/goldbach.html $\endgroup$
    – Penelope
    Mar 14, 2021 at 23:06
  • $\begingroup$ You will find the complete reference here. $\endgroup$ Mar 14, 2021 at 23:07
  • 1
    $\begingroup$ A thread on MO regarding Goldbach's proof of the infinitude of primes: mathoverflow.net/questions/22316/… $\endgroup$ Mar 22, 2021 at 1:10

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