Cross-Posted from Math Stackexchange
Two positive integers $p,q$ and a prime $r$ are given, such that $r>p>q>1$.
I have to show that there exist $n$ such that
$$r|\binom{p^n}{q^n}$$
Should I use Lucas' theorem? I can't solve it.
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5$\begingroup$ May I ask for what purpose do you need it? $\endgroup$– Fedor PetrovCommented Oct 20, 2017 at 11:38
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5$\begingroup$ zhoraster's answer on MSE shows that a natural approach doesn't work: There doesn't have to be a carry in the units digit. If $r=31, p=7, q=4$, then the units digit of $7^n$ is always at least the units digit of $4^n$ base $31$. $\endgroup$– Douglas ZareCommented Oct 20, 2017 at 14:19
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3$\begingroup$ There are no counterexamples among the first 200 primes. For $(p,q,r)=(2,669,1021)$ we need to go to $n=24$. $\endgroup$– Neil StricklandCommented Oct 20, 2017 at 17:57
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3$\begingroup$ Didn't Zhoraster's most recent edit answer the problem completely? $\endgroup$– Pace NielsenCommented Oct 23, 2017 at 13:30
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4$\begingroup$ Imo it is a bad way to cross-post. Firstly, the question were posted here just after the bounty on math.SE had started. You certainly should have waited for the bounty to expire. Secondly, you didn't acknowledge the cross-posting and didn't mention the reasons for it in you post on math.SE. Thirdly, you also started a bounty here, while the one on math.SE was active. You didn't commented on my answer yet, and you didn't acknowledge here that the answer is available. And so on. $\endgroup$– zhorasterCommented Oct 25, 2017 at 6:47
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