关于Diophantine方程x^3+1=3pqy^2

On the Diophantine Equation x^3+1 = 3pqy^2

关于Diophantine方程x^3+1=3pqy^2整数解的情况至今仍未解决。本文主要利用递归数列、同余式、平方剩余以及Pell方程解的性质证明:设素数p≡1(mod 24),素数q=12s^2+1,(s是正奇数),(p/q)=-1,Diophantine方程x^3+1=3pqy^2仅有整数解,即(x,y)=(-1,0)。关键词:Diophantine方程;同余式;平方剩余;

The Diophantine equation x^3 ＋ 1 = 3pqy^2 still remains unresolved. Recurrent sequence, congruence, quadratic residue and Pell equation were used to prove that ： the Diophantine equation x^3 ＋ 1 = 3pqy^2 only has in- teger solution （x ,y） = （ -1,0） when p ,q are primes with p - 1 （mod 24） and, q = 12s2 ＋ 1 （s be positive odd number） , （p/q） = - 1.