Related papers: Irregular primes to two billion

We consider the $k$-error linear complexity of binary sequences derived from Eluer quotients modulo $2p$ ($p>3$ is an odd prime), recently introduced by J. Zhang and C. Zhao. We adopt certain decimal sequences to determine the values of…

We give a precise description of how the class group of a number field measures the failure of unique factorization in its ring of integers. Specifically, following ideas of Kummer, we determine the structure of all irreducible…

We introduce a zeta function counting imaginary quadratic number fields by their class numbers. It is proved that such a function is rational depending only on the eight roots of unity of degrees $1$ and $2$. As a corollary, one gets a…

For any odd prime $p,$ we construct an infinite family of pairs of imaginary quadratic fields $\mathbb{Q}(\sqrt{d}),\mathbb{Q}(\sqrt{d+1})$ whose class numbers are both divisible by $p.$ One of our theorems settles Iizuka's conjecture for…

We prove a higher dimensional generalization of Gross and Zagier's theorem on the factorization of differences of singular moduli. Their result is proved by giving a counting formula for the number of isomorphisms between elliptic curves…

In this paper, it is proved that every sufficiently large even integer can be represented as the sum of two squares of primes, two cubes of primes, two biquadrates of primes and 16 powers of 2. Furthermore, there are at least 5.313% odd…

Let $K$ be a cyclic totally real number field of odd degree over $\mathbb{Q}$ with odd class number, such that every totally positive unit is the square of a unit, and such that $2$ is inert in $K/\mathbb{Q}$. We define a family of number…

We fix a non-zero integer $a$ and consider arithmetic progressions $a \bmod q$, with $q$ varying over a given range. We show that for certain specific values of $a$, the arithmetic progressions $a \bmod q$ contain, on average, significantly…

Let $p>3$ be a prime. Euler numbers $E_{p-3}$ first appeared in H. S. Vandiver's work (1940) in connection with the first case of Fermat Last Theorem. Vandiver proved that $x^p+y^p=z^p$ has no solution for integers $x,y,z$ with…

Let $p$ be an odd prime number. In this article, we study the number of quadratic residues and non-residues modulo $p$ which are multiples of $2$ or $3$ or $4$ and lying in the interval $[1, p-1]$, by applying the Dirichlet's class number…

Fix a rational prime $r \geq 5$. In this article, we study the integer solutions of the generalized Fermat equation of signature $(2p,2q,r)$, namely $x^{2p}+y^{2q}=z^r$, where the primes $p,q \geq 5$ are varying. For each rational prime $r…

We consider the simplest quartic number fields $\mathbb{K}_m$ defined by the irreducible quartic polynomials $$x^4-mx^3-6x^2+mx+1,$$ where $m$ runs over the positive rational integers such that the odd part of $m^2+16$ is squarefree. In…

Let $A$ be a real quadratic order of discriminant $p$ or $4p$ with a prime $p$. In this paper we classify all proper totally imaginary quadratic $A$-orders $B$ with index $w(B)=[B^\times: A^\times]>1$. We also calculate numerical invariants…

Let p be an odd prime. Let F_p^* be the no-null part of the finite field of p elements. Let K=\Q(zeta) be a p-cyclotomic field and O_K be its ring of integers. Let pi be the prime ideal of K lying over p. Let sigma : zeta --> zeta^v be the…

Let K be a number field, and let a be a non-zero element of K. Fix some prime number l. We compute the density of the following set: the primes p of K such that the multiplicative order of the reduction of a modulo p is coprime to l (or,…

We present an efficient algorithm to compute the Euler factor of a genus 2 curve C/Q at an odd prime p that is of bad reduction for C but of good reduction for the Jacobian of C (a prime of ``almost good'' reduction). Our approach is based…

In this paper we study some structure properties of primitive weird numbers in terms of their factorization. We give sufficient conditions to ensure that a positive integer is weird. Two algorithms for generating weird numbers having a…

Let p be an odd prime. Let K = Q(zeta) be the p-cyclotomic field. Let v be any primitive root mod p. Let sigma be a Q-isomorphism of K. Let P(sigma) = sigma^{p-2}v^{-(p-2)}+ ... + sigma v^{-1} +1 \in Z[G] where 1 \leq v^n \leq p-1 is a…

Let $k\geq 3$ and $n\geq 3$ be odd integers, and let $m\geq 0$ be any integer. For a prime number $\ell$, we prove that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{\ell^{2m}-2k^n})$ is either divisible by $n$ or by a…

We address the general mathematical problem of computing the inverse $p$-th root of a given matrix in an efficient way. A new method to construct iteration functions that allow calculating arbitrary $p$-th roots and their inverses of…