2
votes
1answer
541 views

A formula combining Euler $\phi$ and $\gcd$

Let us fix a natural number $N>1$ and $a_1, \ldots, a_n$ natural numbers satisfying $0 \leq a_i < N$, with the property that $1+ \sum a_i$ is divisible by $N$. Let $\phi$ be the Euler totient ...
4
votes
0answers
283 views

Generalization of Tamarkin’s ARO 1993, final round, problem 10/8: part II

Let us use the notations of my previous question about Tamarkin's problem. Let $\ell\in\left\lbrace 0,1,...,p\right\rbrace$. An element $f\in \mathbb Z^{\mathbb Z}$ is said to be ...
29
votes
2answers
2k views

Generalization of Tamarkin's ARO 1993, final round, problem 10/8: still a conjecture?

This is from the category "problems I cannot believe that are still open". But then again, I don't know whether it is still open; it seems to have escaped the attention of most number theorists and ...
17
votes
3answers
1k views

A binomial sum is divisible by p^2

This is a question I have since longer time, but I have absolutely no idea how to proceed on it. Let $p>3$ be a prime. Prove that ...