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### A system of simultaneous congruences arising from trinomial exponential sums

Journal de Théorie des Nombres de Bordeaux

For a prime $p$ and positive integers $\ell <k<h<p$ with $d=\left(h,k,\ell ,p-1\right)$, we show that $M$, the number of simultaneous solutions $x,y,z,w$ in ${ℤ}_{p}^{*}$ to ${x}^{h}+{y}^{h}={z}^{h}+{w}^{h}$, ${x}^{k}+{y}^{k}={z}^{k}+{w}^{k}$, ${x}^{\ell }+{y}^{\ell }={z}^{\ell }+{w}^{\ell }$, satisfies$M\le 3{d}^{2}{\left(p-1\right)}^{2}+25hk\ell \left(p-1\right).$When $hk\ell =o\left(p{d}^{2}\right)$ we obtain a precise asymptotic count on $M$. This leads to the new twisted exponential sum bound$\left|\sum _{x=1}^{p-1}\chi \left(x\right){e}^{2\pi if\left(x\right)/p}\right|\le {3}^{\frac{1}{4}}{d}^{\frac{1}{2}}{p}^{\frac{7}{8}}+\sqrt{5}{\left(hk\ell \right)}^{\frac{1}{4}}{p}^{\frac{5}{8}},$for trinomials $f=a{x}^{h}+b{x}^{k}+c{x}^{\ell }$, and to results on the average size of such sums.

### Algebraically solvable problems: describing polynomials as equivalent to explicit solutions.

The Electronic Journal of Combinatorics [electronic only]

Acta Arithmetica

Acta Arithmetica

Integers

### Counting with Gauss' sums.

Divulgaciones Matemáticas

Acta Arithmetica

Acta Arithmetica

### Équations diophantiennes modulo ${p}^{2}$

Colloquium Mathematicae

Integers

### Identities for direct decomposability of congruences can be written in two variables

Commentationes Mathematicae Universitatis Carolinae

### Incomplete exponential sums and incomplete residue systems for congruences

Czechoslovak Mathematical Journal

### Lower limit for the number of sets of solutions of xe + ye + ze ... 0 (mod p).

Journal für die reine und angewandte Mathematik

Integers

### Note on the number of solutions of the congruence $f\left({x}_{1},{x}_{2},\cdots ,{x}_{n}\right)\equiv 0\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}p\right)$

Mathematica Slovaca

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

Colloquium Mathematicae

For every finite Abelian group Γ and for all $g,a₁,...,{a}_{k}\in \Gamma$, if there exists a solution of the equation ${\sum }_{i=1}^{k}{a}_{i}{x}_{i}=g$ in non-negative integers ${x}_{i}\le {b}_{i}$, where ${b}_{i}$ are positive integers, then the number of such solutions is estimated from below in the best possible way.

Acta Arithmetica

### On a linear homogeneous congruence

Colloquium Mathematicae

The number of solutions of the congruence $a₁x₁+\cdots +{a}_{k}{x}_{k}\equiv 0\left(modn\right)$ in the box $0\le {x}_{i}\le {b}_{i}$ is estimated from below in the best possible way, provided for all i,j either $\left({a}_{i},n\right)|\left({a}_{j},n\right)$ or $\left({a}_{j},n\right)|\left({a}_{i},n\right)$ or $n|\left[{a}_{i},{a}_{j}\right]$.

### On a system of equations with primes

Journal de Théorie des Nombres de Bordeaux

Given an integer $n\ge 3$, let ${u}_{1},...,{u}_{n}$ be pairwise coprime integers $\ge 2$, $𝒟$ a family of nonempty proper subsets of $\left\{1,...,n\right\}$ with “enough” elements, and $\epsilon$ a function $𝒟\to \left\{±1\right\}$. Does there exist at least one prime $q$ such that $q$ divides ${\prod }_{i\in I}{u}_{i}-\epsilon \left(I\right)$ for some $I\in 𝒟$, but it does not divide ${u}_{1}\cdots {u}_{n}$? We answer this question in the positive when the ${u}_{i}$ are prime powers and $\epsilon$ and $𝒟$ are subjected to certain restrictions.We use the result to prove that, if ${\epsilon }_{0}\in \left\{±1\right\}$ and $A$ is a set of three or more primes that contains all prime divisors of any number of the form ${\prod }_{p\in B}p-{\epsilon }_{0}$ for...

### On Equations y² = xⁿ+k in a Finite Field

Bulletin of the Polish Academy of Sciences. Mathematics

Solutions of the equations y² = xⁿ+k (n = 3,4) in a finite field are given almost explicitly in terms of k.

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