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Suppose we write $s(n,k)$ for the Stirling numbers of the first kind. When $p$ is a prime number, I'm interested in knowing when $s(p^a, k)$ is divisible by $p^a$. So:

What is known about $s(p^a,k)$ mod $p^a$ ?

I have tried to ask google about this, but the literature is confusing (there are some congruences that are proved, but it seems difficult to figure out whether this one is among them). I thought that someone here might know the answer from the top of their head.

(Note that I would really like the answer to be "you need to work it out yourself", for it seems like a fun problem. Unfortunately someone else may have done it already.)

Thanks !

Pierre

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  • $\begingroup$ Have you come about this one? Tamás Lengyel, On p-adic properties of the Stirling numbers of the first kind, Journal of Number Theory Volume 148, March 2015, Pages 73–94. sciencedirect.com/science/article/pii/S0022314X14003102 $\endgroup$
    – Wolfgang
    Commented Apr 14, 2015 at 16:40
  • $\begingroup$ @Wolfgang: I'll check this tomorrow when I'm back at work, but it seems to be just what I needed! $\endgroup$
    – Pierre
    Commented Apr 14, 2015 at 20:10
  • $\begingroup$ @Wolfgang: thanks, it's a great reference. It proves a number of useful results, and I take it that what it doesn't prove is not known... if you write this as an answer I'll accept it. $\endgroup$
    – Pierre
    Commented Apr 15, 2015 at 13:09

1 Answer 1

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As said in my comment, this article gives some partial results: Tamás Lengyel, On p-adic properties of the Stirling numbers of the first kind, Journal of Number Theory Volume 148, March 2015, Pages 73–94.

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