On the Greatest Common Divisor of Binomial Coefficients ${n \choose q}, {n \choose 2q}, {n \choose 3q}, \dots$

Every binomial coefficient aficionado knows that the greatest common divisor of the binomial coefficients $(\binom{n}{1}), (\binom{n}{2}), \dots, (\binom{n}{n - 1})$ equals $p$ if $n = p^{i}$ for some $i > 0$ and equals 1 otherwise. It is less well known that the greatest common divisor of the binomial coefficients $(\binom{2 n}{2}), (\binom{2 n}{4}), \dots, (\binom{2 n}{2 n - 2})$ equals (a certain power of 2 times) the product of all odd primes $p$ such that $2 n = p^{i} + p^{j}$ for some $0\le i\le j$. This note gives a concise proof of a tidy generalization of these facts.

On the Greatest Common Divisor of Binomial Coefficients $(\binom{n}{q}), (\binom{n}{2 q}), (\binom{n}{3 q}), \dots$

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On the Greatest Common Divisor of Binomial Coefficients $(\binom{n}{q}), (\binom{n}{2 q}), (\binom{n}{3 q}), \dots$

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On the Greatest Common Divisor of Binomial Coefficients (nq),(n2q),(n3q),…(qn​),(2qn​),(3qn​),…

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On the Greatest Common Divisor of Binomial Coefficients (nq),(n2q),(n3q),…(qn​),(2qn​),(3qn​),…

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On the Greatest Common Divisor of Binomial Coefficients $(\binom{n}{q}), (\binom{n}{2 q}), (\binom{n}{3 q}), \dots$

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On the Greatest Common Divisor of Binomial Coefficients $(\binom{n}{q}), (\binom{n}{2 q}), (\binom{n}{3 q}), \dots$

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