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

Let $m\in {\mathbb{Z}}_{>0}$ and a,q ∈ ℚ. Denote by ${}_m(a,q)$ the set of rational numbers d such that a, a + q, ..., a + (m-1)q form an arithmetic progression in the Edwards curve $E_d:x²+y²=1+dx²y²$. We study the set ${}_m(a,q)$ and we parametrize it by the rational points of an algebraic curve.

We show that, with suitable modification, the upper bound estimates of Stolt for the fundamental integer solutions of the Diophantine equation Au²+Buv+Cv²=N, where A>0, N≠0 and B²-4AC is positive and nonsquare, in fact characterize the fundamental solutions. As a corollary, we get a corresponding result for the equation u²-dv²=N, where d is positive and nonsquare, in which case the upper bound estimates were obtained by Nagell and Chebyshev.

We consider systems of equations of the form $x_i+x_j=x_k$ and $x_i·x_j=x_k$, which have finitely many integer solutions, proposed by A. Tyszka. For such a system we construct a slightly larger one with much more solutions than the given one.

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.

Let $F_n$ denote the $n^{th}$ term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation $F_1^p+2F_2^p+\cdots +k{F}_k^p=F_n^q$ in the positive integers $k$ and $n$, where $p$ and $q$ are given positive integers. A complete solution is given if the exponents are included in the set $\{1,2\}$. Based on the specific cases we could solve, and a computer search with $p,q,k\le 100$ we conjecture that beside the trivial solutions only $F_8=F_1+2F_2+3F_3+4F_4$, $F_4^2=F_1+2F_2+3F_3$, and $F_4^3=F_1^3+2F_2^3+3F_3^3$ satisfy the title equation.

In this study, we determine when the Diophantine equation $x^2-kxy+y^2-2^n=0$ has an infinite number of positive integer solutions $x$ and $y$ for $0\le n\le 10.$ Moreover, we give all positive integer solutions of the same equation for $0\le n\le 10$ in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation $x^2-kxy+y^2-2^n=0$.

This paper is concerned with the estimation of the number of integer solutions to exponential equations in several variables, over function fields. We develop a method which sometimes allows to replace known exponential bounds with polynomial ones. More generally, we prove a counting result (Thm. 1) on the integer points where given exponential terms become linearly dependent over the constant field. Several applications are given to equations (Cor. 1) and to the estimation of the number of equal...