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What is the simplest proof one can think of (perhaps as close as possible to the standard Euclidean one) that there are infinitely many primes of the form $36k-1$?

If you consider something like $36p_1p_2\ldots p_n - 1$ then there would a prime factor of the form $4m-1$, but it might not be congruent to $(-1)$ mod $9$.

More generally, my aim is to give an elementary proof that there is an infinite sequence $a_1, a_2, \ldots$ of positive integers such that GCD$(36a_i - 1, 36a_j - 1) = 1$ for $i\neq j$.

Any help appreciated!

DesmondMiles
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    Showing that infinitely many primes occur in that sequence may not have such an elementary proof, but your question about the gcd does, as your can use the chinese remainder theorem to construct such a sequence inductively – Mike Daas May 27 '21 at 22:29
  • For an elementary proof, see https://doi.org/10.4169/amer.math.monthly.122.01.48 – lhf May 28 '21 at 00:51
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    See also https://math.stackexchange.com/a/3346456/589 – lhf May 28 '21 at 00:53
  • @Ihf is it possible to access the article freely? – Mohammadreza May 29 '21 at 09:30

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