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.

The goal of this paper is to outline the proof of a conjecture of Gelfond [6] (1968) in a recent work in collaboration with Christian Mauduit [11] concerning the sum of digits of prime numbers, reflecting the lecture given in Edinburgh at the Journées Arithmétiques 2007.

Grosswald’s conjecture is that g(p), the least primitive root modulo p, satisfies g(p) ≤ √p - 2 for all p > 409. We make progress towards this conjecture by proving that g(p) ≤ √p -2 for all $409<p<2.5\times {10}^{15}$ and for all $p>3.38\times {10}^{71}$.

A set $𝒮=\{x_1,...,x_n\}$ of $n$ distinct positive integers is said to be gcd-closed if $(x_i,x_j)\in 𝒮$ for all $1\le i,j\le n$. Shaofang Hong conjectured in 2002 that for a given positive integer $t$ there is a positive integer $k\left(t\right)$ depending only on $t$, such that if $n\le k\left(t\right)$, then the power LCM matrix $\left({[x_i,x_j]}^t\right)$ defined on any gcd-closed set $𝒮=\{x_1,...,x_n\}$ is nonsingular, but for $n\ge k\left(t\right)+1$, there exists a gcd-closed set $𝒮=\{x_1,...,x_n\}$ such that the power LCM matrix $\left({[x_i,x_j]}^t\right)$ on $𝒮$ is singular. In 1996, Hong proved $k\left(1\right)=7$ and noted $k\left(t\right)\ge 7$ for all $t\ge 2$. This paper develops Hong’s method and provides a new idea to calculate...

W. Sierpiński asked in 1959 (see [4], pp. 200-201, cf. [2]) whether there exist infinitely many positive integers not of the form n - φ(n), where φ is the Euler function. We answer this question in the affirmative by proving Theorem. None of the numbers $2^k·509203$ (k = 1, 2,...) is of the form n - φ(n).

In the first part, we assign to each positive integer $n$ a digraph $\Gamma (n,5),$ whose set of vertices consists of elements of the ring ${\mathbb{Z}}_n=\{0,1,\cdots ,n-1\}$ with the addition and the multiplication operations modulo $n,$ and for which there is a directed edge from $a$ to $b$ if and only if $a^5\equiv b(modn)$. Associated with $\Gamma (n,5)$ are two disjoint subdigraphs: ${\Gamma }_1(n,5)$ and ${\Gamma }_2(n,5)$ whose union is $\Gamma (n,5).$ The vertices of ${\Gamma }_1(n,5)$ are coprime to $n,$ and the vertices of ${\Gamma }_2(n,5)$ are not coprime to $n.$ In this part, we study the structure of $\Gamma (n,5)$ in detail. In the second part, we investigate the zero-divisor...