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Arithmetical Definability over Finite Structures
, 2002
"... Arithmetical definability has been extensively studied over the natural numbers. In this paper, we take up the study of arithmetical definability overfinite structures, motivated by the correspondence between uniform AC and FO(PLUS;TIMES). We prove finite analogs of three classic results in arithmet ..."
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Arithmetical definability has been extensively studied over the natural numbers. In this paper, we take up the study of arithmetical definability overfinite structures, motivated by the correspondence between uniform AC and FO(PLUS;TIMES). We prove finite analogs of three classic results in arithmetical definability, namely that < and TIMES can firstorder define PLUS, that < and DIVIDES can firstorder define TIMES, and that < and COPRIME can firstorder define TIMES.
Short presentations for alternating and symmetric groups
"... We construct two kinds of presentations for the alternating and symmetric groups of degree n: the first are on two generators in which the number of relations is O(log n) and the presentation length is O(log 2 n); the second have a bounded number of generators and relations and length O(log n). ..."
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We construct two kinds of presentations for the alternating and symmetric groups of degree n: the first are on two generators in which the number of relations is O(log n) and the presentation length is O(log 2 n); the second have a bounded number of generators and relations and length O(log n).
An upper bound in Goldbach's problem
 Math. Comp
, 1993
"... : It is clear that the number of distinct representations of a number n as the sum of two primes is at most the number of primes in the interval [n=2; n \Gamma 2]. We show that 210 is the largest value of n for which this upper bound is attained. 1. Introduction. In 1742 Christian Goldbach wrote, ..."
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: It is clear that the number of distinct representations of a number n as the sum of two primes is at most the number of primes in the interval [n=2; n \Gamma 2]. We show that 210 is the largest value of n for which this upper bound is attained. 1. Introduction. In 1742 Christian Goldbach wrote, in a letter to Euler, that on the evidence of extensive computations he was convinced that every integer exceeding 6 was the sum of three primes. Euler replied that if an even number 2n + 2 is so represented then one of those primes must be even and thus 2, so that every even number 2n, greater than 2, can be represented as the sum of two primes; it is easy to see that this conjecture implies Goldbach's original proposal, and it has widely become known as Goldbach's conjecture. Although still unresolved, Goldbach's conjecture is widely believed to be true. It has now been verified for every even integer up to 2 \Theta 10 10 (in [3]), and there are many interesting partial results worthy ...
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"... presentations for alternating and symmetric groups J.N. Bray, M.D.E. Conder, C.R. LeedhamGreen, and E.A. O’Brien We derive new families of presentations (by generators and relations) for the alternating and symmetric groups of finite degree n. These include presentations of length that are linear i ..."
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presentations for alternating and symmetric groups J.N. Bray, M.D.E. Conder, C.R. LeedhamGreen, and E.A. O’Brien We derive new families of presentations (by generators and relations) for the alternating and symmetric groups of finite degree n. These include presentations of length that are linear in log n, and 2generator presentations with a bounded number of relations independent of n. 1