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A well known theorem of Dirichlet says: if $a,b$ are relatively prime positive integers, then there are infinitely many primes of the form $an+b$.

The original proof by Dirichlet (possibly) uses analytical methods. Also, in special cases, such as for $a=4$ and $b=1$ or $3$, the theorem can be proved in elementary fashion, without using analysis.

However, I heard, that there is no proof of Dirichlet theorem which avoids analysis.

Another interesting theorem in number theory is "$n!$ is never a perfect square" whose proof uses again analytical methods (see this).

Question. What are the other theorems in number theory (other than F.L.T) which are easy to state, but whose proof necessarily depends on analytical methods?

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  • I think that Minkowski's theorem deal with this matter! – Deliasaghi Sep 18 '15 at 11:22
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    The proof of $n!$ never being a perfect square doesn't require analysis. Bertrand's postulate (what is meant by Chebyshev's theorem) has an elementary proof in this paper (by Erdös, section $2$). Dirichlet's theorem for primes of the form $nk+1$ can be easily proved using cyclotomic polynomials without analysis (proofs can be found in many places, e.g. here). – user236182 Sep 18 '15 at 11:23
  • I don't know Minkowski's "theorem" in number theory. Can you state? – Groups Sep 18 '15 at 11:28
  • see https://en.wikipedia.org/wiki/Geometry_of_numbers – Deliasaghi Sep 18 '15 at 11:33

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