关于不定方程x~3-1=181y~2

设p为素数且p≡1(mod 6).关于不定方程x~3-1=py~2的求解是数论的重要研究课题之一.研究p=181时不定方程x~3-1=py~2的可解性问题.利用递归数列,同余式,Pell方程解的性质证明了不定方程x~3-1=181y~2仅有整数解(x,y)=(1,0).

On the diophantine equation x~3-1=181y~2

Let be a prime with p≡1(mod 6).Solving the diophantine equation x~3-1=py~2 is one of the important topics in the number theory.The solvability of the diophantine equation x~3-1=py~2 with p=181 is investigated.By using the recurrent sequence,congruence and some properties of the solutions to Pell equation,it is proved that the diophantine equation x~3-1=181y~2 has only integer solution(x,y)=(1,0).