For any positive integer N, we completely determine the structure of rational cuspidal divisor class group X0(N), which is conjecturally equal to torsion subgroup J0(N). More specifically, for a given prime ℓ, construct Zℓ(d) non-trivial d N. Also, compute order linear equivalence and show that ℓ-primary X0(N) isomorphic direct sum cyclic subgroups generated by classes Zℓ(d).

Twenty-five years ago, W. M. Snyder extended the notion of a repunit Rn to one in which for some positive integer b, Rn(b) has a b-adic expansion consisting of only ones. He then applied algebraic number theory in order to determine the pairs of integers under which Rn(b) has a prime divisor congruent to 1 modulo n. In this paper, we show how Snyder’s theorem follows from existing theory pertai...

Let be a prime, a positive integer, = , and a divisor of ( 1). We derive lower bounds on the linear complexity over the residue class ring of a ( -periodic) sequence representing the residues modulo of the discrete logarithm in . Moreover, we investigate a sequence over representing the values of a certain polynomial over introduced by Mullen and White which can be identified with the discrete ...

We show that, in a restricted range, the divisor function of integers in residue classes modulo a prime follows a Gaussian distribution, and a similar result for Hecke eigenvalues of classical holomorphic cusp forms. Furthermore, we obtain the joint distribution of these arithmetic functions in two related residue classes. These results follow from asymptotic evaluations of the relevant moments...

In 1857 Bouniakowsky [6] made a conjecture concerning prime values of polynomials that would, for instance, imply that x + 1 is prime for infinitely many integers x. Let ƒ (x) be a polynomial with integer coefficients and define the fixed divisor of ƒ, written d(ƒ), as the largest integer d such that d divides f(x) for all integers x. Bouniakowsky conjectured that if f(x) is nonconstant and irr...

For any positive integer n, the Pseudo-Smarandache function Z(n) is defined as the smallest positive integer k such that n | k(k + 1) 2 . That is, Z(n) = min { k : n| + 1) 2 } . The main purpose of this paper is using the elementary methods to study the mean value properties of p(n) Z(n) , and give a sharper asymptotic formula for it, where p(n) denotes the smallest prime divisor of n.

The integrality of the Kontsevich integral and perturbative invariants is discussed. We show that the denominator of the degree n part of the Kontsevich integral of any knot or link is a divisor of (2!3! . . . n!)(n + 1)!. We also show that the denominator of of the degree n part of the universal perturbative invariant of homology 3-spheres is not divisible by any prime greater than 2n + 1.