Let $p>3$ be a prime, and let $q_p\left(2\right)=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base $2$. In this note we prove that $$\sum _{k=1}^{p-1}\frac{1}{k·2^k}\equiv q_p\left(2\right)-\frac{p{q}_p{\left(2\right)}^2}{2}+\frac{p^2{q}_p{\left(2\right)}^3}{3}-\frac{7}{48}{p}^2{B}_{p-3}(mod{p}^3),$$ which is a generalization of a congruence due to Z. H. Sun. Our proof is based on certain combinatorial identities and congruences for some alternating harmonic sums. Combining the above congruence with two congruences by Z. H. Sun, we show that $$q_p{\left(2\right)}^3\equiv -3\sum _{k=1}^{p-1}\frac{2^k}{k^3}+\frac{7}{16}\sum _{k=1}^{(p-1)/2}\frac{1}{k^3}(modp),$$ which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum ${\sum }_{k=1}^{p-1}1/(k^2·2^k)$ modulo $p^2$ that also generalizes a...

Let $\ell \ne p$ be two different prime numbers, let $F$ be a local non archimedean field of residual characteristic $p$, and let ${\overline{𝐐}}_{\ell },{\overline{𝐙}}_{\ell },{\overline{𝐅}}_{\ell }$ be an algebraic closure of the field of $\ell$-adic numbers ${𝐐}_{\ell }$, the ring of integers of ${\overline{𝐐}}_{\ell }$, the residual field of ${\overline{𝐙}}_{\ell }$. We proved the existence and the unicity of a Langlands local correspondence over ${\overline{𝐅}}_{\ell }$ for all $n\ge 2$, compatible with the reduction modulo $\ell$ in [V5], without using $L$ and $ϵ$factors of pairs. We conjecture that the Langlands local correspondence over ${\overline{𝐐}}_{\ell }$ respects congruences modulo $\ell$ between...

The purpose of this paper is to interpret the results of Jakubec and his collaborators on congruences of Ankeny-Artin-Chowla type for cyclic totally real fields as an elementary algebraic version of the p-adic class number formula modulo powers of p. We show how to generalize the previous results to congruences modulo arbitrary powers $p^t$ and to equalities in the p-adic completion ${\mathbb{Q}}_p$ of the field of rational numbers ℚ. Additional connections to the Gross-Koblitz formula and explicit congruences for...

It is classical that a natural number n is congruent iff the rank of ℚ -points on Eₙ: y² = x³-n²x is positive. In this paper, following Tada (2001), we consider generalised congruent numbers. We extend the above classical criterion to several infinite families of real number fields.

This paper investigates the system of equations $$x^2+a{y}^m=z_1^2,x^2-a{y}^m=z_2^2$$ in positive integers $x$, $y$, $z_1$, $z_2$, where $a$ and $m$ are positive integers with $m\ge 3$. In case of $m=2$ we would obtain the classical problem of congruent numbers. We provide a procedure to solve the simultaneous equations above for a class of the coefficient $a$ with the condition $gcd(x,z_1)=gcd(x,z_2)=gcd(z_1,z_2)=1$. Further, under same condition, we even prove a finiteness theorem for arbitrary nonzero $a$.

Ce travail est essentiellement consacré à la construction d’exemples effectifs de couples $(\alpha ,\beta )$ de nombres réels à constantes de Markov finies, tels que $1,\alpha$ et $\beta$ soient $𝐙$-linéairement indépendants, et satisfaisant à la conjecture de Littlewood.

Pour un nombre premier impair $p$ et une extension abélienne $K/k$ de corps de nombres totalement réels, nous utilisons la Conjecture Principale Équivariante démontrée par Ritter et Weiss (modulo la nullité de l’invariant ${\mu }_p$) pour calculer l’idéal de Fitting d’un certain module d’Iwasawa sur l’algèbre complète ${\mathbb{Z}}_p\left[\left[G_{\infty }\right]\right],$ où $G_{\infty }=Gal(K_{\infty }/k)$ et $K_{\infty }$ est la ${\mathbb{Z}}_p$-extension cyclotomique de $K$. Par descente, nous en déduisons la $p$-partie de la version cohomologique de la conjecture de Coates-Sinnott, ainsi qu’une forme faible de la $p$-partie...