In this paper we have found out all integer solutions of the Diophanine equation in the title, which was proposed by Mordell in 1969 and reproposed by Guy in 1981.

• 1969年,Mordell提出求解不定方程6y~2=(x+1)(x~2-x+6),本文解决了这个问题,求得了这个方程的全部整数解。