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

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 $(⁎)$ $x^2=2^{2a+2}{p}^{2n}-2^{a+2}{p}^{m+n}+1$, $p\in P,x,z,m,n\in N$ 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 $2^a=(g+1)v_{t+1}-(g-1)v_t$ and $p^n=v_{t+1}+v_t$, where $v_r=({\alpha }^r-{\overline{\alpha }}^r)/(\alpha -\overline{\alpha })$ for any integer r, $\alpha =2g+\sqrt{4g^2+1}$ and $\overline{\alpha }=2g-\sqrt{4g^2+1}$. 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 $x<C\left(p\right)$, where $C\left(p\right)$ is an effectively computable constant depending only on p.