关于不定方程ax^4+x^3+x-a=Dy^2

On the Diophanfine Equation ax^4-x^3+x-a=Dy^2

设1+4a^2为素数或为2个形如8m+1型的素数的乘积,本文讨论不定方程 ax^4+x^3+x-a=Dy^2,其中 a、D 为整数,a≠0,D>0且无平方因子,主要结果为当 D 不被2或4k+1型的素数整除时,且 a=±2(mod 8),则该不定方程无解,若 a<0,仅当a=Db^2(b≥1)时有唯一解,若a>0,则当 a=Db^2(b≥1)时有解 x=2a。

The author makes a solution to the diophantine equation ax^4+x^3+x-a=Dy^2 with two integers a≠ 0,D>0,and D square-free. When 1+4a^2 is a prime or a product of primes of the form 8m+1,then (1)There is no integer solution of the equation if D can not be integrally divided by 2 or by a prime of the form 4k+1 and a≡+2(mod 8). (2)There is a unique solution of the equation if a<0,and only when a=Db^2(b≥1). If a<0,and when a=Db^2(b≥1),then x=2a.