关于不定方程x^(2)-3y^(4)=p(p=13,37,61,73)

On the Diophantine Equation x^(2)-3y^(4)=p(p=13,37,61,73)

运用递归序列、同余式以及平方剩余的有关性质,证明了以下结论:(1)不定方程x^(2)-3y^(4)=13仅有正整数解(x,y)=(4,1)和(16,3);(2)不定方程x^(2)-3y^(4)=37没有正整数解;(3)不定方程x^(2)-3y^(4)=61仅有正整数解(x,y)=(8,1)和(44,5);(4)不定方程x^(2)-3y^(4)=73仅有正整数解(x,y)=(11,2)和(29,4)。

Recurrent sequence,congruence and quadratic residue are used to prove the following conclusions.①Diophantine equation x^(2)-3y^(4)=13 has only positive integer solutions(x,y)=(4,1),(16,3);②Diophantine equation x^(2)-3y^(4)=37 has no positive integer solution;③Diophantine equation x^(2)-3y^(4)=61 has only positive integer solutions(x,y)=(8,1),(44,5);④Diophantine equations x^(2)-3y^(4)=73 has only positive integer solutions(x,y)=(11,2),(29,4).