关于丢番图方程y2=px(x2+2)

数学季刊 ›› 2009, Vol. 24 ›› Issue (4): 499-503.

关于丢番图方程y²=px(x²+2)

Department of Mathematics, Northwest University

收稿日期:2006-06-06 出版日期:2009-12-30 发布日期:2023-06-16
作者简介:WANG Xiao-ying(1964-), female, native of Xianyang, Shannxi, an associate professor of Northwest University, Ph.D., engages in analytic number theory.
基金资助:
Supported by the Natural Science Foundation of Shaanxi Province(2009JM1006)；

On the Diophantine Equation y²=px(x²+2)

Department of Mathematics, Northwest University

Received:2006-06-06 Online:2009-12-30 Published:2023-06-16
About author:WANG Xiao-ying(1964-), female, native of Xianyang, Shannxi, an associate professor of Northwest University, Ph.D., engages in analytic number theory.
Supported by:
Supported by the Natural Science Foundation of Shaanxi Province(2009JM1006)；

摘要/Abstract

摘要： For any fixed odd prime p, let N(p) denote the number of positive integer solutions (x,y) of the equation y²=px(x²+2). In this paper, using some properties of binary quartic Diophantine equations, we prove that if p≡5 or 7(mod 8), then N(p)=0; ifp≡1(mod 8), then N(p)≤1; ifp>3 and p≡3(mod 8), then N(p)≤2.

关键词: cubic and quartic Diophantine equation, number of solutions, upper bound

Abstract: For any fixed odd prime p, let N(p) denote the number of positive integer solutions (x,y) of the equation y²=px(x²+2). In this paper, using some properties of binary quartic Diophantine equations, we prove that if p≡5 or 7(mod 8), then N(p)=0; ifp≡1(mod 8), then N(p)≤1; ifp>3 and p≡3(mod 8), then N(p)≤2.

Key words: cubic and quartic Diophantine equation, number of solutions, upper bound

中图分类号:

O156.7

王晓瑛. 关于丢番图方程y²=px(x²+2)[J]. 数学季刊, 2009, 24(4): 499-503.

WANG Xiao-Ying. On the Diophantine Equation y²=px(x²+2)[J]. Chinese Quarterly Journal of Mathematics, 2009, 24(4): 499-503.

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关于丢番图方程y²=px(x²+2)

On the Diophantine Equation y²=px(x²+2)

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