Contributions to Ma's conjecture concerning abelian difference sets with multiplier −1 (I)

IF 0.9 2区 数学 Q2 MATHEMATICS
Yasutsugu Fujita , Maohua Le
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引用次数: 0

Abstract

Let N, P be the sets of all positive integers and odd primes, respectively. In 1991, when studying the existence of abelian difference sets with multiplier −1, S.-L. Ma [14] conjectured that the equation () x2=22a+2p2n2a+2pm+n+1, pP,x,z,m,nN has only one solution (p,x,a,m,n)=(5,49,3,2,1). This is a far from solved problem that has been poorly known for so long. In this paper, using some elementary methods, we first prove that if (p,x,a,m,n) is a solution of () with m=2n, then there exist an odd positive integer g and a positive integer t which make 2a=(g+1)vt+1(g1)vt and pn=vt+1+vt, where vr=(αrα¯r)/(αα¯) for any integer r, α=2g+4g2+1 and α¯=2g4g2+1. Then, we obtain certain properties of the positive integers t and g. Finally, we comprehensively apply some classical results from transcendental number theory and Diophantine equations to prove that, for any fixed odd prime p, all solutions (x,a,m,n) of () with m=2n satisfy x<C(p), where C(p) is an effectively computable constant depending only on p.
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来源期刊
CiteScore
2.90
自引率
9.10%
发文量
94
审稿时长
12 months
期刊介绍: The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.
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