We find all elliptic curves defined over Q that have a rational point of order N, N ≥ 4, and whose conductor is of the form pq, where p, q are two distinct primes, a, b are two positive integers. In particular, we prove that Szpiro’s conjecture holds for these elliptic curves.

We consider binary abelian codes of length pq where p and q are prime rational integers under some restrictive hypotheses. In this case, we determine the idempotents generating minimal codes and either the respective weights or bounds of these weights. We give examples showing that these bounds are attained in some cases.

We consider a special type of integer partitions in which the parts of the form pq, for some relatively prime integers p and q, are restricted by divisibility conditions. We investigate the problems of generating and encoding those partitions and give some estimates for several partition functions.

Let N = pq be an RSA modulus where p, q are large primes of the same bitsize and φ(N) = (p − 1)(q − 1). We study the class of the public exponents e for which there exist integers X, Y , Z satisfying

We consider which integers g can occur as the genus and of a curve defined over a field of characteristic p which admits an automorphism of degree pq, where p and q are distinct primes. This investigation leads us to consider a certain family of three-dimensional Frobenius problems and prove explicit formulas giving their solution in many cases. Required Publisher's Statement This article was p...

Magic rectangles are well-known for their very interesting and entertaining combinatorics. Such magic rectangles have also been used in designing experiments. In a magic rectangle, the integers 1 to pq are arranged in an array of p rows and q columns so that each row adds to the same total P and each column to the same total Q. In the present paper we provide a systematic method for constructin...

1. The E.A. (Euclidean Algorithm) holds for a quadratic field R(2/.m)? when, for any two integers a, ,G in R(~/m.) with /3 + 0, a third integer y in B(~/w) can be so det,ermined that (1) 1 N(a-/3y) / < j N(P) (or \ N(a/fi-y) 1 < 1. The existence of the E .A. is undecided in the following cases; p, q denoting primes : y I. m=p= 13+24n (,n> 1); II. m=p= 1+&L (n> 7); III. m =pq wit#h 23 = q-3 or p...

A (p, q)-ary chain is a special type of chain partition of integers with parts of the form pq for some fixed integers p and q. In this note, we are interested in the maximal weight of such partitions when their parts are distinct and cannot exceed a given bound m. Characterizing the cases where the greedy choice fails, we prove that this maximal weight is, as a function of m, asymptotically ind...

The signature of a labelled tree (and hence of its prefix-closed branch language) is the sequence of the degrees of the nodes of the tree in the breadth-first traversal. In a previous work, we have characterised the signatures of the regular languages. Here, the trees and languages that have the simplest possible signatures, namely the periodic ones, are characterised as the sets of representat...

We show that all but 4489 integers n with 4 < n ≤ 4 ·1030 cannot occur as the order of a circulant Hadamard matrix. Our algorithm allows us to search 10000 times farther than prior efforts, while substantially reducing memory requirements. The principal improvement over prior methods involves the incorporation of a separate search for double Wieferich prime pairs {p, q}, which have the property...