椭圆曲线y^(2)=(x-6)(x^(2)+6x+m)的整数点

Integral Points on Elliptic Curve y^(2)=(x-6)(x^(2)+6x+m)

设m=30s^(2)-7,其中s是使6s^(2)+13及15s^(2)-8为奇素数的正奇数,结合初等数论方法及二元四次丢番图方程的结论,证明了椭圆曲线y^(2)=(x-6)(x^(2)+6x+m)除整数点(x,y)=(6,0)外无其他非平凡整数点。

Let m=30s^(2)-7,where s is a positive odd number such that 6s^(2)+13 and 15s^(2)-8 are odd primes.Combined with the method of elementary number theory and the conclusion of the binary quadratic Diophantine equation,it is proved that the elliptic curve y^(2)=(x-6)(x^(2)+6x+m)has no other non-trivial integer points except the integer point(x,y)=(6,0).