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Subjects
11-XX Number theory
11Dxx Diophantine equations
11D79 Congruences in many variables

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Neither $\prod_{k = 1}^{n} (4 k 2 + 1)$ nor $\prod_{k = 1}^{n} (2 k (k - 1) + 1)$ is a perfect square.

Fang, Jin-Hui (2009)

Integers

Note on the number of solutions of the congruence $f (x_{1}, x_{2}, \dots, x_{n}) \equiv 0 (mod p)$

Stanislav Jakubec (1994)

Mathematica Slovaca

Number of solutions in a box of a linear equation in an Abelian group

Maciej Zakarczemny (2016)

Colloquium Mathematicae

For every finite Abelian group Γ and for all $g, a ₁, . . ., a_{k} \in Γ$ , if there exists a solution of the equation $\sum_{i = 1}^{k} a_{i} x_{i} = g$ in non-negative integers $x_{i} \leq b_{i}$ , where $b_{i}$ are positive integers, then the number of such solutions is estimated from below in the best possible way.

Number of solutions in a box of a linear homogeneous equation in an Abelian group

Maciej Zakarczemny (2012)

Acta Arithmetica

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