与马猜想有关的一类不定方程

On Some Equations Related to Ma’s Conjecture

设p是奇素数,b.t.r∈N.1992年,马少麟猜想丢番图方程x^(2)=2^(2b)+^(2)p^(2t)-2^(2b)+^(2)p^(t+r)+1有唯一的正整数解(x.b.p.t.r)=(49,3,5,1,2),并且证明了这个猜想蕴含McFarland关于乘子为-1的阿贝尔差集的猜想.在[Ma S L,MaFarland'conjecture on Abelian difference sets with multiplier-1[J].Designs,Codes and Cryptography,1992,1:321-332.]中,马少麟证明了:若t≥r,则丢番图方程x^(2)=2^(2b)+^(2)p^(2t)-2^(2b)+^(2)p^(t+r)+1没有正整数解.本文证明了:若α>1是奇数,t≥r,那么丢番图方程x^(2)=2^(2b)+^(2)p^(2t)-2^(2b)+^(2)p^(t+r)+1的正整数解由t=r=1,x+α√2^(b+2)(2^(b)-1)=(2^(b+1)-1+√2^(b+2)(2^(b)-1)^(n)给出,其中n为奇数.作者也证明了:若p是奇素数,则(x,b,p,t,r)=(7,3,5,1,2)是丢番图方程x^(4)=2^(2b)+^(2)p^(2t)-2^(2b)+^(2)p^(t+r)+1的唯一正整数解.

Let p be an odd prime and b,t,r ∈ N.In 1992,Ma conjectured that(x,b,p,t,r)=(49,3,5,1,2)is the only positive integer solution of equation x^(2)=2^(2b)+^(2)p^(2t)-2^(2b)+^(2)p^(t+r)+1.And Ma proved that the conjecture implies McFarland's conjecture on Abelian difference sets with multiplier-1.In[Ma S L,MaFarland'conjecture on Abelian difference sets with multiplier-1[J].Designs,Codes and Cryptography,1992,1:321-332.],Ma proved that e-quation x^(2)=2^(2b)+^(2)p^(2t)-2^(2b)+^(2)p^(t+r)+1 had no positive integer solution if t≥r.In the present paper,the authors prove that the positive integer solutions of Diophantine e-quation x^(2)=2^(2b)+^(2)p^(2t)-2^(2b)+^(2)p^(t+r)+1 with a is an odd>1 and t≥r are given by t=r=1 and x+α√2^(b+2)(2^(b)-1)=(2^(b+1)-1+√2^(b+2)(2^(b)-1)^(n)for some odd positive integer n.They also prove that the only positive integer solution of Diophantine equation x^(4)=2^(2b)+^(2)p^(2t)-2^(2b)+^(2)p^(t+r)+1with p is an odd prime and x,b,t,r∈ N is given by(x,b,p,t,r)=(7,3,5,1,2).