There n variables x1,x2,...,xn represented as X, n equations whose coefficient matrix (n*n) is represented as A, and this system looks like this: AX = B (mod k) Initially I was t …

Fix an odd prime $p$. Let $\alpha = (\alpha_0,\dots,\alpha_k)$ be a solution to the congruence $\sum_{i=0}^{k} \alpha_i^2 \equiv x \mod p$. Now consider the number $N_\alpha$ of so …

I need a reference for the result which gives the number of solutions to the congruence $mx^2+ny^2 \equiv k\pmod{p}$. This result seems to be something that would be discussed in …

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) …

Suppose we have an invertible matrix q in a finite subgroup $Q$ of $Gl(n,\mathbb Z)$, the group of all invertible integer matrices. Now I want to find all $x\; mod\; \mathbb Z^n$ f …

permutation is given by f(i) = 2i if i<=n and 2(i-n)-1 if i>n where i denotes the position of cards. eg pack of cards (1,2,3,4)-->(3,1,4,2) Basically trying to find the multip …

Dear All, I need to find the result of $1+ 2^x + 2^{2x} + 2^{3x} +\dots+2^{nx} \pmod {2^{x-3}-7 }$. Is there a short way of finding it instead of calculating $\pmod{ 2^{x-3}- …

Let $\sigma_{11}(n)$ denote the sum of the 11th powers of the positive integral divisors of the positive integer n. Let $\tau(n)$ denote Ramanujan's tau function, which is the coef …

One of the recent questions, in fact the answer to it, reminded me about the binomial sequence $$ a_n=\sum_{k=0}^n{\binom{n}{k}}^2{\binom{n+k}{k}}^2, \qquad n=0,1,2,\dots, $$ of th …