Volume 12, Issue 6

Abstract. Define to be the complexity of n , the smallest number of ones needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that for all n . Define the defect of n , denoted by , to be ; in this paper we present a method for classifying all n with for a given r . From this, we derive several consequences. We prove that for with m and k not both zero, and present a method that can, with more computation, potentially prove the same for larger m . Furthermore, defining to be the number of n with and , we prove that , allowing us to conclude that the values of can be arbitrarily large.

Abstract. Here, we show that is the only solution in positive integers of the Diophantine equation where is the m th Fibonacci number.

Abstract. We classify all integer squares (and, more generally, q -th powers for certain values of q ) whose ternary expansions contain at most three digits. Our results follow from Padé approximants to the binomial function, considered 3-adically.

Abstract. We consider the second of Mullin's sequences of prime numbers related to Euclid's proof that there are infinitely many primes. We show in particular that it omits infinitely many primes, confirming a conjecture of Cox and van der Poorten.

Abstract. We continue the analysis of higher and multiple Mahler measures using log-sine integrals as started in “Log-sine evaluations of Mahler measures” and “Special values of generalized log-sine integrals” by two of the authors. This motivates a detailed study of various multiple polylogarithms and worked examples are given. Our techniques enable the reduction of several multiple Mahler measures, and supply an easy proof of two conjectures by Boyd.

Abstract. An idea used in the characterization of even perfect numbers is used, first, to derive new necessary conditions for the existence of an odd perfect number and, second, to show that there are no even 3-perfect numbers of the form , where M is odd and squarefree and , besides the six known examples.

Abstract. We prove that an odd number n is an Euler pseudoprime for exactly one half of the admissible bases if and only if n is a special Carmichael number, that is, for every invertible .

Abstract. This paper presents all known information on incongruent restricted disjoint covering systems with only odd moduli. The results in this paper formed part of the author's doctoral dissertation completed at Macquarie University.

Abstract. The main aim of this paper is to describe a procedure for calculating the number of cubes that have coordinates in the set {0,1,...,n}. For this purpose we continue and, at the same time, revise some of the work begun in a sequence of papers about equilateral triangles and regular tetrahedra all having integer coordinates for their vertices. We adapt the code that was included in a paper by the first author and was used to calculate the number of regular tetrahedra with vertices in {0,1,...,n} 3 . The idea is based on the theoretical results obtained by the first author with A. Markov. We then extend the sequence A098928 in the Online Encyclopedia of Integer Sequences to the first one hundred terms.

Abstract. In 1960, Sierpiński proved that there exist infinitely many odd positive rational integers k such that is composite in for all . Any such integer k is now known as a Sierpiński number, and the smallest value of k produced by Sierpiński's proof is . In 1962, John Selfridge showed that is also a Sierpiński number, and he conjectured that this value of k is the smallest Sierpiński number. This conjecture, however, is still unresolved today. In this article, we investigate the analogous problem in the ring of integers of each imaginary field having class number one. More precisely, for each , with , that has unique factorization, we determine all , with minimal odd norm larger than 1, such that is composite in for all . We call these numbers Selfridge numbers in honor of John Selfridge.

Abstract. Let A and M be nonempty sets of positive integers. A partition of the positive integer n with parts in A and multiplicities in M is a representation of n in the form where for all , and for only finitely many  a . Denote by the number of partitions of n with parts in A and multiplicities in  M . It is proved that there exist infinite sets A and M of positive integers whose partition function has weakly superpolynomial but not superpolynomial growth. The counting function of the set A is . It is also proved that must have at least weakly superpolynomial growth if M is infinite and .

Abstract. Suppose that a is a positive non-square integer and we wish to multiply a subset of to form a square so that a is used and we minimize the largest number used. Now add the requirement that we must use at most f factors in forming the square. Erdős, Malouf, Selfridge, and Szekeres conjectured that if , then using factors results in a smaller maximum than using factors. Their paper on this subject reduces the conjecture to the case that a is a term in a certain sequence , where grows almost as fast as . We give theoretical and computational results which establish the result for and for several infinite classes that comprise a positive proportion of the subscripts N .

Abstract. One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the literature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contributions in the survey on `elliptic Artin' are due to Alina Cojocaru. Wojciec Gajda wrote a section on `Artin for K-theory of number fields,' and Hester Graves (together with me) on `Artin's conjecture and Euclidean domains.'

Abstract. We discuss a relative of the perfect numbers for which it is possible to prove that there are infinitely many examples. Call a natural number n prime-perfect if n and share the same set of distinct prime divisors. For example, all even perfect numbers are prime-perfect. We show that the count of prime-perfect numbers in satisfies estimates of the form as . We also discuss the analogous problem for the Euler function. Letting denote the number of for which n and share the same set of prime factors, we show that as , We conclude by discussing some related problems posed by Harborth and Cohen.

Abstract. We show how to obtain the solutions of families of systems of two Pell equations; these families are parameterized by the prime numbers.

Abstract. We searched the Cunningham tables for new algebraic factorizations similar to those discovered by Aurifeuille. A naive search would have been too slow. We accelerated it enough to make it feasible. Many interesting results were found.

Abstract. A sequence of rational integers is said to be a divisibility sequence if whenever . If the divisibility sequence also satisfies a linear recurrence relation of order k , it is said to be a linear divisibility sequence. The best known example of a linear divisibility sequence of order 2 is the Lucas sequence , one particular instance of which is the famous Fibonacci sequence. In their extension of the Lucas functions to order 4 linear recursions, Williams and Guy showed that the order 4 analog of can have no more than two ranks of apparition for a given prime p and frequently has two such ranks, unlike the situation for , which can only have one rank of apparition. In this paper we investigate the problem of finding those sequences which have only one rank of apparition for any prime p .

Abstract. Due to a technical error, the paper “Sum-product estimates applied to Waring's problem over finite fields” by Todd Cochrane does not appear in this volume but was published as article A68 of Volume 11. We apologize for this error.