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Noticed that my previous two questions:

  1. Are there infinitely many $n$ where the (number of non coprime pairs) $-$ (number of coprime pairs) is 3?
  2. Euler's totient function and primes for even and odd numbers.

seem related and might possibly be combined to prove the infinitude of primes.

Theorem

There are infinitely many primes.

Proof

From $(1)$, there are infinitely many even $n$ such that $$ \varphi(n) = \frac{n-2}{2} $$ From $(2)$, if $$ \varphi(n) = \frac{n-2}{2}, \text{ then } \frac{n}{2} \text{ is prime.} $$

Since there are infinitely many such $n$, it follows that there are infinitely many primes. $\blacksquare$

Question

Is this proof valid? Can the results from my previous questions be combined in this way to prove the infinitude of primes?

vengy
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    Well, you'd need to check carefully that you never used the infinitude of the primes in proving those two claims. For instance, in the accepted solution to $(1)$, the author appears to rely precisely on the fact that there are infinitely many primes. – lulu Aug 13 '24 at 00:01
  • Ah, the circular reasoning. Thanks. – vengy Aug 13 '24 at 00:02
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    It's easy to prove there are infinitely many primes via the totient function though. After all, if $N$ was the product of all the primes then, absurdly, $\varphi(N)=1$. See, e.g., this question – lulu Aug 13 '24 at 00:03
  • @lulu Isn't this proof more or less a copy of the classical Eudlid proof ? – Peter Aug 13 '24 at 07:06
  • @Peter. Sure, only harder since you need to argue that $\varphi(N)=1$ isn't possible. But the OP appears to want a proof using the totient. – lulu Aug 13 '24 at 08:44
  • For another slant on lulu's point, see my answer to this question – Keith Backman Aug 13 '24 at 18:49

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