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 $(a_i,n)|(a_j,n)$ or $(a_j,n)|(a_i,n)$ or $n\left|\right[a_i,a_j]$.

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 $\{1,...,n\}$ with “enough” elements, and $\epsilon$ a function $𝒟\to \{\pm 1\}$. 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 ${\epsilon }_0\in \{\pm 1\}$ 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...

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

This paper is concerned with non-trivial solvability in $p$-adic integers of systems of additive forms. Assuming that the congruence equation $a{x}^k+b{y}^k+c{z}^k\equiv d\left(modp\right)$ has a solution with $xyz\not\equiv 0\left(modp\right)$ we have proved that any system of $R$ additive forms of degree $k$ with at least $2·3^{R-1}·k+1$ variables, has always non-trivial $p$-adic solutions, provided $p\nmid k$. The assumption of the solubility of the above congruence equation is guaranteed, for example, if $p\>k^4$.

We consider an equation of the type$$a_1^{}{x}_1^2+\cdots +a_n^{}{x}_n^2=b{x}_1\cdots x_n$$over the finite field ${𝔽}_q={𝔽}_{p^s}$. Carlitz obtained formulas for the number of solutions to this equation when $n=3$ and when $n=4$ and $q\equiv 3(mod4)$. In our earlier papers, we found formulas for the number of solutions when $d=gcd(n-2,(q-1)/2)=1$ or $2$ or $4$; and when $d\>1$ and $-1$ is a power of $p$ modulo $2d$. In this paper, we obtain formulas for the number of solutions when $d=2^t$, $t\ge 3$, $p\equiv 3\text{or}5(mod8)$ or $p\equiv 9(mod16)$. For general case, we derive lower bounds for the number of solutions.

We show that the intersection of the images of two polynomial maps on a given interval is sparse. More precisely, we prove the following. Let $f\left(x\right),g\left(x\right){\in }_p\left[x\right]$ be polynomials of degrees d and e with d ≥ e ≥ 2. Suppose M ∈ ℤ satisfies $p^{1/E(1+\kappa /(1-\kappa )}>M>p^{\epsilon }$, where E = e(e+1)/2 and κ = (1/d - 1/d²) (E-1)/E + ε. Assume f(x)-g(y) is absolutely irreducible. Then $\left|f\left([0,M]\right)\cap g\left([0,M]\right)\right|\lesssim M^{1-\epsilon }$.