If one takes the following definition: "A period is a number that can be expressed as an integral of an algebraic function over an algebraic domain", how could one go about determining that a certain number, say $\exp(\pi)$ (but I'm not interested in this concrete example), is not a period?

Since I can think of only one way how one could prove that a number is a period - by constructing an integral with the aforementioned properties - the possible way of proving the converse seems impossible for me. But I am missing something, am I not?

UPD: the context is that I was reading paper "Periods" by Maxim Kontsevich and Don Zagier and didn't find any information about how to prove that a number is not a period.

  • $\begingroup$ Yes, indeed, the source is missing, and quite some context. $\endgroup$
    – user436658
    Commented Dec 11, 2020 at 20:06
  • $\begingroup$ I've heard in a seminar that noncomputable numbers are counterexamples, but I am unfamiliar with the proof. $\endgroup$
    – KReiser
    Commented Dec 11, 2020 at 22:07
  • $\begingroup$ I'm pretty sure that non-computable numbers are counterexamples — given the definition of a period one can in principle compute it to any degree of accuracy using any of the normal techniques, and in fact I believe it's even possible to get upper and lower bounds by using interval arithmetic; that's enough to get a 'computation' of the value of the period by any reasonable definition thereof I know. $\endgroup$ Commented Dec 12, 2020 at 6:20
  • 1
    $\begingroup$ Concretely, I believe this is a Hard Problem — AFAIK all the results about various classic numbers not being periods are conjectural. $\endgroup$ Commented Dec 12, 2020 at 6:23
  • $\begingroup$ @StevenStadnicki so the impression that one simply looks for an integral for long enough time and then comes up with a conjecture that the number is not a period? $\endgroup$
    – user108687
    Commented Dec 12, 2020 at 6:34


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