Reference request: gcd of binomial coefficients

Question

Let $n > 1$. What is a good reference for the following fact?

$$\mathrm{gcd} \left\{\binom{n}{k} : 0 < k < n \right\} = \begin{cases} p & n = p^m \text{ is a prime power} \\ 1 & \text{else}\end{cases}$$

It has been discussed on math.SE here. I am not looking for a proof, I am looking for a citable reference which also includes a proof. This paper gives a reference to a paper from 1909 (B. Ram, Common factors of n!/m!(n−m)!) which I cannot access. So I am also looking for a reference which is easy to find. It does not have to be original. Ideally it would be a modern number theory textbook.

In a paper, citing a SE answer for a well-known result is bad practice. — Martin Brandenburg, Nov 27 '22 at 11:13
A few articles you may try: https://www.sciencedirect.com/science/article/pii/0022314X85900137 https://www.sciencedirect.com/science/article/pii/S1631073X16301054 https://www.jstor.org/stable/10.4169/amer.math.monthly.124.4.353 — Jean-Claude Arbaut, Nov 27 '22 at 11:19
@Jean-ClaudeArbaut I already did. See also the text in my question. — Martin Brandenburg, Nov 27 '22 at 11:39
Not sure I understand: I read your question, and I understand there is an article you can't find. While looking for it I found these ones. But if you already found them and they are of no use, then I think I'm not able to help more. I may check in a couple of textbooks, but it's the kind of result I expect to find among exercises. — Jean-Claude Arbaut, Nov 27 '22 at 11:42

score 1 · Accepted Answer · answered Oct 03 '23 at 18:30

1

Just to put this one off the unanswered list ...

B. Ram, Common factors of n!/m!(n−m)!, J. Indian Math. Club (Madras) 1 (1909) 39-43

And thanks again @Jean-Claude Arbault for sending me the article!

answered Oct 03 '23 at 18:30

Martin Brandenburg

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Reference request: gcd of binomial coefficients

1 Answers1