方程y^2=nx(x^2±1)的正整数解数的上界(英文)

The Upper Bound of Positive Integer Solution Number of the Equation y^2=nx(x^2±1)

设n是无平方因子正整数.本文利用二次和四次Diophantine方程解数的结果,讨论了方程y^2=nx(x^2±1)的正整数解个数的上界,证明了该方程至多有2~w(n)个正整数解(x,y),其中w(n)是n的不同素因数的个数.

Let n be a positive integer with square free. In this paper, using the results of the number of solutions of quadratic and quartic diophantine equations, we discuss the upper bound for the number of positive integer solutions of the equation y2 = nx（x2 4- 1）. We prove that the equation has at most 2^ω（n） positive integer solutions （x, y）, where w（n） is the number of distinct prime divisors of n.