Articles by Mercier and the author [10, 11] discuss the situation that S1, S2 are the images of N under certain (monotonic) polynomial functions p1, p2 with integer coefficients. In the present paper, we will consider (in fact in a more general context) the case that one or both of S1, S2 is equal to the set B = BQ(i) consisting of those natural numbers which can be written as a sum of two inte...

This paper establishes that every positive-definite matrix can be written as a positive linear combination of outer products of integer-valued vectors whose entries are bounded by the geometric mean of the condition number and the dimension of the matrix.

We present a method for updating certain hierarchical factorizations for solving linear integral equations with elliptic kernels. In particular, given a factorization corresponding to some initial geometry or material parameters, we can locally perturb the geometry or coefficients and update the initial factorization to reflect this change with asymptotic complexity that is polylogarithmic in t...

In this paper we address the problem of matrix factorization on compressively-sampled measurements which are obtained by random projections. While this approach improves the scalability of matrix factorization, its performance is not satisfactory. We present a matrix co-factorization method where compressed measurements and a small number of uncompressed measurements are jointly decomposed, sha...

We describe how we reached a new factoring milestone by completing the first special number field sieve factorization of a number having more than 1024 bits, namely the Mersenne number 2 − 1. Although this factorization is orders of magnitude ‘easier’ than a factorization of a 1024-bit RSA modulus is believed to be, the methods we used to obtain our result shed new light on the feasibility of t...

Let n = p1 1 · · · pr r be the prime factorization of a positive integer n. Define the excess of n to be (α1−1)+ · · ·+(αr −1), which is the difference between the total multiplicity α1 + · · · + αr and the number of distinct primes in the factorization. An integer with excess 0 is also said to be square-free. Let Ek denote the set of positive integers of excess k, k = 0, 1, 2, . . .. Rényi pro...

Introduction. A long-standing problem in group theory is to determine the number of non-isomorphic groups of a given order. The inverse problem–determining the orders for which there are a given number of groups–has received considerably less attention. In this note, we will give a characterization of those positive integers n for which there exist exactly 2 distinct groups of order n (up to is...

Factoring large integers into primes is one of the most important and most difficult problems of computational number theory (the twin problem is primality testing [13]). Trial division, Fermat's algorithm [1], [3], [8], Pollard's p-\ method [6], Williams' p + \ method [11], Lenstra's elliptic curve method (ECM) [5], Pomerance's quadratic sieve (QS) [7], [10], and Pollard's number field sieve (...