Let $a$, $b$, $c$, $r$ be positive integers such that $a^2+b^2=c^r$, $min(a,b,c,r)>1$, $gcd(a,b)=1,a$ is even and $r$ is odd. In this paper we prove that if $b\equiv 3\left(mod4\right)$ and either $b$ or $c$ is an odd prime power, then the equation $x^2+b^y=c^z$ has only the positive integer solution $(x,y,z)=(a,2,r)$ with $min(y,z)>1$.

In the paper we discuss the following type congruences: $$\left(\genfrac{}{}{0pt}{}{n{p}^k}{m{p}^k}\right)\equiv \left(\genfrac{}{}{0pt}{}{m}{n}\right)(mod{p}^r),$$ where $p$ is a prime, $n$, $m$, $k$ and $r$ are various positive integers with $n\ge m\ge 1$, $k\ge 1$ and $r\ge 1$. Given positive integers $k$ and $r$, denote by $W(k,r)$ the set of all primes $p$ such that the above congruence holds for every pair of integers $n\ge m\ge 1$. Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets $W(k,r)$ and inclusion relations between them for various values $k$ and $r$. In particular, we prove that $W(k+i,r)=W(k-1,r)$ for all $k\ge 2$, $i\ge 0$ and...

A classical theorem of M. Fried [2] asserts that if non-zero integers $\beta ₁,...,{\beta }_l$ have the property that for each prime number p there exists a quadratic residue ${\beta }_j$ mod p then a certain product of an odd number of them is a square. We provide generalizations for power residues of degree n in two cases: 1) n is a prime, 2) n is a power of an odd prime. The proofs involve some combinatorial properties of finite Abelian groups and arithmetic results of [3].

Let $a_{d-1},\cdots ,a_0\in \mathbb{Z}$, where $d\in \mathbb{N}$ and $a_0\ne 0$, and let $X={\left(x_n\right)}_{n=1}^{\infty }$ be a sequence of integers given by the linear recurrence $x_{n+d}=a_{d-1}{x}_{n+d-1}+\cdots +a_0{x}_n$ for $n=1,2,3,\cdots$. We show that there are a prime number $p$ and $d$ integers $x_1,\cdots ,x_d$ such that no element of the sequence $X={\left(x_n\right)}_{n=1}^{\infty }$ defined by the above linear recurrence is divisible by $p$. Furthermore, for any nonnegative integer $s$ there is a prime number $p\ge 3$ and $d$ integers $x_1,\cdots ,x_d$ such that every element of the sequence $X={\left(x_n\right)}_{n=1}^{\infty }$ defined as above modulo $p$ belongs to the set $\{s+1,s+2,\cdots ,p-s-1\}$.

We show that if p ≠ 5 is a prime, then the numbers $1/p\left(\genfrac{}{}{0pt}{}{p}{m₁,...,m_t}\right)|t\ge 1,m_i\ge 0fori=1,...,tand{\sum }_{i=1}^t{m}_i=p$ cover all the nonzero residue classes modulo p.