For a ternary quadratic form over the rational numbers, we characterize the set of rational numbers represented by that form over the rational numbers. Consequently, we reprove the classical fact that any positive definite integral ternary quadratic form must fail to represent infinitely many positive integers over the rational numbers. Our proof uses only the quadratic reciprocity law and the Hasse-Minkowski theorem, and is elementary.

Let t, b be mutually prime positive integers. We say that the residue class t mod b is basic if there exists n such that tn ≡ -1 mod b; otherwise t is not basic. In this paper we relate the basic character of t mod b to the quadratic character of t modulo the prime factors of b. If all prime factors p of b satisfy p ≡ 3 mod 4, then t is basic mod b if t is a quadratic non-residue mod p for all such p; and t is not basic mod b if t is a quadratic residue mod p for all such p. If, for all prime factors...

A power digraph, denoted by $G(n,k)$, is a directed graph with ${\mathbb{Z}}_n=\{0,1,\cdots ,n-1\}$ as the set of vertices and $E=\{(a,b):a^k\equiv b(modn)\}$ as the edge set. In this paper we extend the work done by Lawrence Somer and Michal Křížek: On a connection of number theory with graph theory, Czech. Math. J. 54 (2004), 465–485, and Lawrence Somer and Michal Křížek: Structure of digraphs associated with quadratic congruences with composite moduli, Discrete Math. 306 (2006), 2174–2185. The heights of the vertices and the components of $G(n,k)$ for $n\ge 1$ and $k\ge 2$ are determined....

Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of integers m and n. Let p be a prime of the form 4k+1 and p = c²+d² with c,d ∈ ℤ, $d=2^{r}d₀$ and c ≡ d₀ ≡ 1 (mod 4). In the paper we determine ${(b+\surd (b²+4^{\alpha })/2)}^{(p-1)/4)}\left(modp\right)$ for p = x²+(b²+4α)y² (b,x,y ∈ ℤ, 2∤b), and ${(2a+\surd 4a²+1)}^{(p-1)/4}\left(modp\right)$ for p = x²+(4a²+1)y² (a,x,y∈ℤ) on the condition that (c,x+d) = 1 or (d₀,x+c) = 1. As applications we obtain the congruence for $U_{(p-1)/4}\left(modp\right)$ and the criterion for $p|U_{(p-1)/8}$ (if p ≡ 1 (mod 8)), where Uₙ is the Lucas sequence given by U₀ = 0, U₁ = 1 and $U_{n+1}=bUₙ+U_{n-1}\left(n\ge 1\right)$, and b ≢...

For any two positive integers $n$ and $k\ge 2$, let $G(n,k)$ be a digraph whose set of vertices is $\{0,1,...,n-1\}$ and such that there is a directed edge from a vertex $a$ to a vertex $b$ if $a^k\equiv b(modn)$. Let $n={\prod }_{i=1}^r{p}_i^{e_i}$ be the prime factorization of $n$. Let $P$ be the set of all primes dividing $n$ and let $P_1,P_2\subseteq P$ be such that $P_1\cup P_2=P$ and $P_1\cap P_2=\varnothing$. A fundamental constituent of $G(n,k)$, denoted by $G_{P_2}^{*}(n,k)$, is a subdigraph of $G(n,k)$ induced on the set of vertices which are multiples of ${\prod }_{p_i\in P_2}{p}_i$ and are relatively prime to all primes $q\in P_1$. L. Somer and M. Křížek proved that the trees attached to all cycle...