A theorem of Siegel asserts that

If $\beta>0$ and $2^\beta,\,3^\beta,\,5^\beta$ are integers, then $\beta$ is an integer.

The following result is a beautiful consequence of this theorem

If $\beta$ is a positive number such that $1^\beta,\,2^\beta,\,3^\beta,\,\dotsc$ are integers, then $\beta$ is itself is an integer.

I'm looking for a proof of this result.

Note. This result appeared as a problem in the 1972 Putnam Prize competition, and not one of more than 2000 university student competitors gave a solution; the solution, though not hard, could well elude even a professional mathematician for several hours (or days).

  • $\begingroup$ I fixed typos in the title. Is it OK ? $\endgroup$ Apr 26 '15 at 14:40
  • 2
    $\begingroup$ This is a restatement of Problem A6 from the 1971 Putnam; a solution appears here. $\endgroup$
    – vadim123
    Apr 26 '15 at 14:43
  • $\begingroup$ We may notice that it is a pretty straightforward consequence of Baker's theorem (en.wikipedia.org/wiki/Baker%27s_theorem). $\endgroup$ Apr 26 '15 at 14:44
  • $\begingroup$ @vadim123: oh, now I see. The solution through finite differences is very nice and tricky! $\endgroup$ Apr 26 '15 at 14:47
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    $\begingroup$ There is a solution for the problem with using finite differences, however the generalisation of the Siegel theorem for two bases is open. $\endgroup$
    – k1.M
    Apr 27 '15 at 8:17

Well, k1.M, here an extension of important and beautiful Siegel Theorem (I did not know). It is clear $\beta$ must be integer or irrational and by another theorem of Siegel $\beta$ must be trascendental. I guess the statement goes for any three coprime integer x, y, z the triple 2, 3, 5 having been chosen for smallest.Thus the true spirit of the statement of Siegel would be "all trascendental as exponent of an integer can gives an integer at most twice".

  • $\begingroup$ The sentence written in bold letters is the same Teorem of Siegel taking 2,3,5 and is just to say it is VERY PROFOUND despite some persons might think. $\endgroup$
    – Piquito
    Apr 26 '15 at 23:52

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