1
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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

6
$\begingroup$

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.

$\endgroup$
4
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.