Conditions for the Solubility of the Diophantine Equation(a~n-1)((a+1)~n-1)=x~2

Let a be a positive integer with a1,and let a≡λ(mod 2),where λ∈{0,1}.Let f(a)=ord 2(a-λ) denote the degree of 2 in the factorization of a-λ.In this paper,using some elementary number theories,we prove that if the equation(an-1)((a+1)n-1)=x2 has positive integer solutions(n,x),then a satisfies the following conditions:(i) f(a)=2r,where r is a positive integer with r1.(ii) Every odd prime divisor p of a+1 satisfies p≡±1(mod 8).