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In Erdős, Graham, Ruzsa, and Straus - On the prime factors of $\binom{2n}n$, at the beginning of the proof of theorem 1, they consider the case where $\log p$ and $\log q$ are commensurable numbers (...

For some prime $p$, it is a standard approach based on Kummer's criterion to bound the number of positive integers $n<X$ for some parameter $X$, such that $p\nmid \binom{2n}{n}$. However, if we ...

Take $$B(l,n)=\binom{n+l}{n}$$ and $\mathcal P(t)=\{p\mbox{ prime}:p|t\}$. What is the cardinality of $\mathcal P(B(l,n))$? What is minimum cardinality of $L\subseteq\{1,\dots,n\}$ such that $$\...

For the Dirichlet character $\chi(a)=(\frac a3)$ (which is the Legendre symbol), we have $$L(2,\chi)=\sum_{n=1}^\infty\frac{(\frac n3)}{n^2}=0.781302412896486296867187429624\ldots.$$ Note that this ...

The sequence https://oeis.org/A287326 - is Binomial distributed triangular array, that shows us necessary items to expand perfect cube $n^3$. Summation of $n$-th row of Triangle A287326 from $0$ to $n-...

If $r\in\Bbb Z_{\geq0}$ and $m$ is odd then let $2^\ell\mid\binom{m}{2^r}$ and $2^{\ell+1}\nmid\binom{m}{2^r}$. Is there a way to find if $\ell$ is even or odd without computing $\binom{m}{2^r}$ (...