# Computing Non-zero End Digits of Large Factorials [closed]

Any large factorial will have a number of zero behind it, and one could write an expression to compute the number of trailing zeros, but how would one go about computing the non-zero end digits?

E.g. compute first 5 non-zero end digits of (10^12)!

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## closed as too localized by Igor Rivin, Matthew Daws, Kevin O'Bryant, Felipe Voloch, Emil JeřábekNov 9 '11 at 18:18

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Not a research level question, voting to close. –  Igor Rivin Nov 9 '11 at 14:13
Indeed not research level, though it does lead to the $p$-adic Gamma function (here with $p=5$). –  Noam D. Elkies Nov 9 '11 at 14:18
Computing the first 5 non-zero end digits of (10^12)! is Project Euler problem 160. Clearly this kind of cheating shouldn't be encouraged. –  Guntram Nov 9 '11 at 16:16

## 1 Answer

This will be the least complete answer I've ever posted, but perhaps somebody else will be able to add to it. Not a complete answer (because I don't find any paper online), but I've heard Jean-Marc Deshouillers speak on his (with others, but I don't recall with whom) proof that the last non-zero base-$p$ digit of $n!$ forms a $q$-automatic sequence. I don't recall the hypothesis on $p$, nor the connection between $p$ and $q$.

But this answers (to a very limited extent) the real math-research level question hidden in this homework question. Or at least it would, if I could recall any details!

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What does "q-automatic" mean? –  Igor Rivin Nov 9 '11 at 14:23
@Igor: there are many equivalent definitions, and the Allouche/Shallit book "Automatic Sequence" is beautifully written; I can recommend it easily for any "tourist" in the area. The wikipedia link is somewhat spartan: en.wikipedia.org/wiki/Automatic_sequence –  Kevin O'Bryant Nov 9 '11 at 15:36