Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

It is trivial that there are no polynomial function $P$ with integer coefficients that has the property $P(n)=p_n$ where $p_n$ is the $n$th prime.While it is true that can always construct a smooth function with this property.But what if we require the function to be real analytic ? Does there exists an analytic function $P$ that has the property $P(n)=p_n$ and its taylor series has integer coefficients?

share|improve this question
Taylor series with nontrivial integral coefficients usually have a small radius of convergence. Do you actually mean that the nth coefficient is (a_n)/n! for integers a_n? Or do you want the series expansion around every integer to have integer coefficients? Gerhard "Scratching His Head On This" Paseman, 2012.03.02 –  Gerhard Paseman Mar 2 '12 at 10:56
Specifically, a Taylor series with integer coefficients, infinitely many of which are nonzero, has radius of convergence at most $1$. –  Emil Jeřábek Mar 2 '12 at 11:42
(In case it’s not obvious, it’s easy to find an entire function which does the job if we allow the coefficients to be real.) –  Emil Jeřábek Mar 2 '12 at 12:06
@Paseman I mean a_n is integer, –  nemesiso Mar 2 '12 at 13:39

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.