关于丢番图方程x^3±1=Dy^2

ON THE DIOPHANTINE EQUATIONS x 3±1=Dy 2

对于丢番图方程（１）ｘ３＋１＝Ｄｙ２和（２）ｘ３－１＝Ｄｙ２，本文用初等方法证明了以下结果：①设Ｄ＝２ａ·３·ｐ或２ａ·３·ｐｓｉ＝１ｑｉ，这里ｐ、ｑｉ均是奇素数，ａ＝０或１，ｑｉ≡５（ｍｏｄ６）（ｉ＝１，…，ｓ），ｓ＝１或２，则当ｐ∈｛７，１３，３１，６１，７３，７９，９７｝时，方程（１）除开Ｄ＝３·５·７仅有解（ｘ，ｙ）＝（３１４，５４３）外，无其他的正整数解，而方程（２）除开Ｄ＝３·５·９７仅有解（ｘ，ｙ）＝（４３６６，７５６３）外，无其他的正整数解。②当Ｄ＝ｐ∈｛１３，３１，４３，６１，６７，９７｝时，方程（１）没有正整数解；当Ｄ＝ｐ∈｛１３，１９，３７，４３，６７，９７｝时，方程（２）没有正整数解。

For the Diophantine equations (1) x 3+1=Dy 2 and (2) x 3-1=Dy 2, we have proved the following theorems, by elementary method, in this paper. Theorem 1 Let D=2 a·3·p or 2 a·3·psi=1q i, where a=0 or 1, q i≡5(mod6)(i=1,…,s),1≤s≤2, p∈{7,13,31,61,73,79,97}, then the equation (1) has no positive integer solution (x,y),except 314 3+1=3·5·7·543 2; the equation (2) has no positive integer sotution (x,y), except 4366 3-1=3·5·97·7563 2. Therem 2 Let D=p is prime, P∈{13,31,43,61,67,97}, then the equation (1) has no positive intger solution (x,y). Let D=p is prime, p∈{13,19,37,43,67,97}, then the equation (2) has no positive integer solution (x,y).