We use the properties of $p$-adic integrals and measures to obtain general congruences for Genocchi numbers and polynomials and tangent coefficients. These congruences are analogues of the usual Kummer congruences for Bernoulli numbers, generalize known congruences for Genocchi numbers, and provide new congruences systems for Genocchi polynomials and tangent coefficients.

We generalize the concept of reduced Arakelov divisors and define C-reduced divisors for a given number C ≥ 1. These C-reduced divisors have remarkable properties, similar to the properties of reduced ones. We describe an algorithm to test whether an Arakelov divisor of a real quadratic field F is C-reduced in time polynomial in $log|{\Delta }_F|$ with ${\Delta }_F$ the discriminant of F. Moreover, we give an example of a cubic field for which our algorithm does not work.

Let $L=K\left(\alpha \right)$ be an extension of a number field $K$, where $\alpha$ satisfies the monic irreducible polynomial $P\left(X\right)=X^p-\beta$ of prime degree belonging to ${𝔬}_K\left[X\right]$ (${𝔬}_K$ is the ring of integers of $K$). The purpose of this paper is to study the monogenity of $L$ over $K$ by a simple and practical version of Dedekind’s criterion characterizing the existence of power integral bases over an arbitrary Dedekind ring by using the Gauss valuation and the index ideal. As an illustration, we determine an integral basis of a pure nonic field $L$ with a...

Let $S$ be a subset of ${\mathbf{F}}_q$, the field of $q$ elements and $h\in {\mathbf{F}}_q\left[x\right]$ a polynomial of degree $d\>1$ with no roots in $S$. Consider the group generated by the image of $\{x-s\mid s\in S\}$ in the group of units of the ring ${\mathbf{F}}_q\left[x\right]/\left(h\right)$. In this paper we present a number of lower bounds for the size of this group. Our main motivation is an application to the recent polynomial time primality testing algorithm [AKS]. The bounds have also applications to graph theory and to the bounding of the number of rational points on abelian covers of the projective...

The class numbers h⁺ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows, and methods employing Leopoldt's decomposition of the class number become hard to use when the field extension is not cyclic of prime power. This is why other methods have been developed, which approach the problem from different angles. In this paper we extend one of these methods that was designed for real cyclotomic fields of prime...

We present algorithms for the computation of extreme binary Humbert forms in real quadratic number fields. With these algorithms we are able to compute extreme Humbert forms for the number fields $\mathbb{Q}\left(\sqrt{13}\right)$ and $\mathbb{Q}\left(\sqrt{17}\right)$. Finally we compute the Hermite-Humbert constant for the number field $\mathbb{Q}\left(\sqrt{13}\right)$.

We describe an algorithm due to Gauss, Shanks and Lagarias that, given a non-square integer $D\equiv 0,1$ mod $4$ and the factorization of $D$, computes the structure of the $2$-Sylow subgroup of the class group of the quadratic order of discriminant $D$ in random polynomial time in $log\left|D\right|$.

In this paper, we find all integer solutions $(x,y,n,a,b,c)$ of the equation in the title for non-negative integers $a,b$ and $c$ under the condition that the integers $x$ and $y$ are relatively prime and $n\ge 3$. The proof depends on the famous primitive divisor theorem due to Bilu, Hanrot and Voutier and the computational techniques on some elliptic curves.

In this paper, we find all solutions of the Diophantine equation $x^2+2^{\alpha }{5}^{\beta }{17}^{\gamma }=y^n$ in positive integers $x,y\ge 1$, $\alpha ,\beta ,\gamma ,n\ge 3$ with $gcd(x,y)=1$.

In this article the discrete logarithm problem in degree 0 class groups of curves over finite fields given by plane models is studied. It is proven that the discrete logarithm problem for non-hyperelliptic curves of genus 3 (given by plane models of degree 4) can be solved in an expected time of $\tilde{O}\left(q\right)$, where $q$ is the cardinality of the ground field. Moreover, it is proven that for every fixed natural number $d\ge 4$ the following holds: We consider the discrete logarithm problem for curves given by plane models...