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Part of a proof that there are infinitely many primes. (I have no special interest in the thing being proved). Page 105 of the 2003 digital printing, proof of Statement 14: (This is part of a proof by contradiction.)

"Consequently q divides both N and N-1 and, hence their difference N - (N-1), i.e. q | 1. However this is impossible since q is prime."

In my self-study, this sounds both ugly and redundant. For me, q cannot divide 1 if q > 1. q being prime is irrelevant.

Nothing divides both N and N-1 other than 1. For example, 2 could divide N and N - 2; 3 could divide N and N - 3. (Fundamental Theorem of Arithmetic)

Why might he have said such a thing? Is it just 1 is not prime, so q is not 1?

analog
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  • The assumption that $q$ is prime was probably needed at an earlier part of the proof. At the stage of the proof you're referring to, all that's needed is $q > 1$, which is true since $q$ is prime. – quasi Apr 04 '21 at 10:27
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    Yes, the writer meant "Since $q$ is prime, $q>1$ and so $q$ can not divide $1$." – lulu Apr 04 '21 at 10:28
  • See here and here and here for more conceptual insight on this and closely proofs. – Bill Dubuque Apr 04 '21 at 12:53
  • Thanks quasi and lulu. It makes more sense now. Thanks Bill for compiling the links. – analog Apr 04 '21 at 22:07

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