On The Diophantine Equation X²=p^2b+2a^2t-p^b+2a^t+rδ+1

Let p be an odd prime and b,t,r∈N.In 1992,Ma conjectured that the only positive integer solution of Diophantine equation x2=22b+2 p2t-2b+2 pt+r+1 is given by(x,p,b,t,r)=(49,3,5,1,2).And Ma proved that the conjecture implies McFarland’s conjecture on Abelian difference sets with multiplier-1.In the paper,Ma proved that the above equation had no positive integer solution if t≥r.This dissertation continues to be generalized to x2=p2b+2a2t-pb+2 t+rδ+1,t≥rδ∈ {±1,±2,±4} and we discuss that if a is an odd>1,t≥r and δ∈｛±1},the Diophantine equation x2-p2b+2a2t-pb+2at+rδ+1 has the necessary and sufficient conditions for the positive integer solutions.And its all positive integer solution is obtained.Also proved the Diophantine equation x2=p2b+2a2t-pb+at rδ+1 has no solution when a is an odd>1,t>r and 8 δ∈ {±2,±4}.