. Monier and Rabin proved that an odd composite can pass the Strong Probable Prime Test for at most 1 4 of the possible bases. In this paper, a probable prime test is developed using quadratic polynomials and the Frobenius automorphism. The test, along with a fixed number of trial divisions, ensures that a composite n will pass for less than 1 7710 of the polynomials x 2 \Gamma bx \Gamma c with i b 2 +4c n j = \Gamma1 and \Gamma \Gammac n \Delta = 1. The running time of the test is asymptotically 3 times that of the Strong Probable Prime Test. x1 Background Perhaps the most common method for determining whether or not a number is prime is the Strong Probable Prime Test. Given an odd integer n, let n = 2 r s + 1 with s odd. Choose a random integer a with 1 a n \Gamma 1. If a s j 1 mod n or a 2 j s j \Gamma1 mod n for some 0 j r \Gamma 1, then n passes the test. An odd prime will pass the test for all a. The test is very fast; it requires no more than (1 +...

Let p; q; r; s be polynomials with integer coecients. This paper presents a fast method, using very little temporary storage, to nd all small integers (a; b; c; d) satisfying p(a)+q(b) = r(c)+s(d). Numerical results include all small solutions to a ; all small solutions to a ; and the smallest positive integer that can be written in 5 ways as a sum of two coprime cubes.

We consider a distributed environment consisting of n processors that need to perform t tasks. We assume that communication is initially unavailable and that processors begin work in isolation. At some unknown point of time an unknown collection of processors may establish communication. Before processors begin communication they execute tasks in the order given by their schedules. Our goal is to schedule work of isolated processors so that when communication is established for the rst time, the number of redundantly executed tasks is controlled. We quantify worst case redundancy as a function of processor advancements through their schedules. In this work we rene and simplify an extant deterministic construction for schedules with n t, and we develop a new analysis of its waste. The new analysis shows that for any pair of schedules, the number of redundant tasks can be controlled for the entire range of t tasks. Our new result is asymptotically optimal: the tails of these schedules are within a 1 +O(n 1 4 ) factor of the lower bound. We also present two new deterministic constructions one for t n, and the other for t n 3=2 , which substantially improve pairwise waste for all prexes of length t= p n, and oer near optimal waste for the tails of the schedules. Finally, we present bounds for waste of any collection of k 2 processors for both deterministic and randomized constructions. 1

this paper is to give a partially expository account of results related to coverings of the integers (defined below) while at the same time making some new observations concerning a related polynomial problem. The polynomial problem we will consider is to determine whether for a given positive integer

One of the main tasks of additive number theory is to examine structural properties of sumsets. For a set A of integers, the sumset lA = A + ···+ A consists of those numbers which can be represented as a sum of l elements of A: lA = {a1 + ···+ al|ai ∈ Ai}. Closely related and equally interesting notion is that of l∗A,whichisthecol lection of numbers which can be represented as a sum of l different elements of A: l ∗ A = {a1 + ···+ al|ai ∈ Ai,ai � = aj}. Among the most well-known results in all of mathematics are Vinogradov’s theorem, which says that 3P (P is the set of primes) contains all sufficiently large odd numbers, and Waring’s conjecture (proved by Hilbert, Hardy and Littlewood, Hua, and many others), which asserts that for any given r, there is a number l such that l∗Nr (Nr denotes the set of rth powers) contains all sufficiently large positive integers (see [29] for an excellent exposition concerning these results). In recent years, a considerable amount of attention has been paid to the study of finite sumsets. Given a finite set A and a positive integer l, the natural analogue

Abstract. This paper reports on new results for the equation m� a k i = n� b k j, i=1 j=1 i.e., equal sums of like powers. Since the 1967 Lander, Parkin and Selfridge survey paper [4], few other numeric results have been published (see Elkies [6] and Ekl [3]). The present paper reports on several new smallest primitive solutions. Further, search limits have been extended in many cases, and tables of solutions are presented. Additionally, new solutions to the same class of problems in distinct integers have been discovered. Introduction. Diophantine equations of the form m�

We de ne a multiplicative arithmetic function D by assigning D(p , when p is a prime and a is a positive integer, and, for n 1, we set D (n)) when k 1. We term fD k=0 the derived sequence of n. We show that all derived sequences of n < 1:5 10 are bounded, and that the density of those n 2 N with bounded derived sequences exceeds 0.996, but we conjecture nonetheless the existence of unbounded sequences. Known bounded derived sequences end (eectively) in cycles of lengths only 1 to 6, and 8, yet the existence of cycles of arbitrary length is conjectured. We prove the existence of derived sequences of arbitrarily many terms without a cycle.

Abstract. A positive integer n is said to be harmonic when the harmonic mean H(n) of its positive divisors is an integer. Ore proved that every perfect number is harmonic. No nontrivial odd harmonic numbers are known. In this article, the list of all harmonic numbers n with H(n) ≤ 300 is given. In particular, such harmonic numbers are all even except 1. 1.

To Peter Hagis, Jr., on the occasion of his 70th birthday Abstract. A natural number n is said to be harmonic when the harmonic mean H(n) of its positive divisors is an integer. These were first introduced almost fifty years ago. In this paper, all harmonic numbers less than 2 × 10 9 are listed, along with some other useful tables, and all harmonic numbers n with H(n) ≤ 13 are determined. 1. Let τ(n) andσ(n) denote the number of positive divisors of a positive integer n, and their sum, respectively. The harmonic mean of these divisors is easily seen to be H(n) = nτ(n) σ(n). Then n is said to be harmonic if H(n) is an integer. Harmonic numbers were first studied by Ore [7], and they remain of interest because of their connection with perfect numbers. Recall that n is perfect if σ(n) =2n; it is easy to show that every perfect number is harmonic. A list of the harmonic numbers less than 2 · 10 9 is given in Table 3, at the end of this paper. This extends the lists of Ore [7] and Garcia [3], which gave all harmonic numbers up to 10 5 and 10 7, respectively. We see that no nontrivial example of an odd harmonic number is known; if it could be proved that in fact there are none, then this would imply the nonexistence of odd perfect numbers. In [4] Guy wrote: “Which values does the harmonic mean take? Presumably not

Factors enabling this discovery are advances in computing power, available workstation memory, and the appropriate choice of optimized algorithms.