A famous unsolved conjecture of P. Erdős and J. L. Selfridge states that there does not exist a covering system {as(mod ns)}s=1 with the moduli n1, . . . , nk odd, distinct and greater than one. In this paper we show that if such a covering system {as(mod ns)}s=1 exists with n1, . . . , nk all square-free, then the least common multiple of n1, . . . , nk has at least 22 prime divisors.

The paper describes a "probabilistic algorithm" for finding a factor of any large composite integer n (the required input is the integer n together with an auxiliary sequence of random numbers). It is proved that the expected number of operations which will be required is 0(exp{ /ftln n In In n)1/2}) for some constant ß > 0. Asymptotically, this algorithm is much faster than any previously anal...

For the integer D = pq of the product of two distinct odd primes, we construct an elliptic curve E2rD : y 2 = x3 − 2rDx over Q, where r is a parameter dependent on the classes of p and q modulo 8, and show, under Birch and Swinnerton-Dyer conjecture, that the elliptic curve has rank one and vp(x([k]Q)) 6= vq(x([k]Q)) for odd k and a generator Q of the free part of E2rD(Q). Thus we can get p or ...

The combinatorial sum of binomial coe cients  n i r (a) := X k⌘i (mod r) ✓ n k ◆ a k has been studied widely in combinatorial number theory, especially when a = 1 and a = 1. In this paper, we connect it with integer factorization for the first time. More precisely, given a composite n, we prove that for any a coprime to n there exists a modulus r such that the combinatorial sum has a nontrivia...

The analysis of preconditioners based on incomplete Cholesky factorization in which the neglected (dropped) components are orthogonal to the approximations being kept is presented. General estimate for the condition number of the preconditioned system is given which only depends on the accuracy of individual approximations. The estimate is further improved if, for instance, only the newly compu...

A classical theorem in number theory due to Euler states that a positive integer z can be written as the sum of two squares if and only if all prime factors q of z, with q ≡ 3 (mod 4), occur with even exponent in the prime factorization of z. One can consider a minor variation of this theorem by not allowing the use of zero as a summand in the representation of z as the sum of two squares. View...

The purpose of this paper is to report the unexpected results that we obtained while experimenting with the multi-large prime variation of the general number field sieve integer factoring algorithm (NFS, cf. [8]). For traditional factoring algorithms that make use of at most two large primes, the completion time can quite accurately be predicted by extrapolating an almost quartic and entirely ‘...

We present a (random) polynomial-time algorithm to generate a randomGaussian integer with the uniform distribution among those with norm at most N , along with its prime factorization. The method generalizes to finding a random ideal in the ring of integers of a quadratic number field together with its prime ideal factorization. We also discuss the analogous problem for higher degree number fie...

using qcd factorization for the hadronic matrix elements, we show that existing data, inparticular the branching ratios br ( b →j/ψk) and br ( b →j/ψπ), can be accounted for in this approach.we analyze the decay b j /k( ) within the framework of qcd factorization. the calculation of therelevant hard-scattering kernels for twist-2 and twist-3 is completed. we calculate this decay in a special...