丢番图方程与同余数方程解
Diophantine Equations and Congruent Number Equation Solutions
摘要：
利用全等数方程$$ C_N:\;\;y^2=x^3-N^2x, $$的非平凡有理解对，构造了有理右（勾股）三角形对，它们的一条公共边和另一边等于相同有理数的平方和与差.我们找到了下列丢番图系统的参数化：\begin{align*} (p^2\pm q^2)^2-a^2 & =\square_{1,2}\,, \\[0.2cm] c^2-(p^2\pm q^2)^2 & =\square_{1,2}\,, \\[0.2cm] a^2+(p^2\pm q^2)^2 & =\square_{1,2}\,, \\[0.2cm] (p^2\pm q^2)^2-a^2 & =(r^2\pm s^2)^2. \end{align*}
英文摘要：
By using pairs of nontrivial rational solutions of congruent number equation $$ C_N:\;\;y^2=x^3-N^2x, $$ constructed are pairs of rational right (Pythagorean) triangles with one common side and the other sides equal to the sum and difference of the squares of the same rational numbers. The parametrizations are found for following Diophantine systems: \begin{align*} (p^2\pm q^2)^2-a^2 & =\square_{1,2}\,, \\[0.2cm] c^2-(p^2\pm q^2)^2 & =\square_{1,2}\,, \\[0.2cm] a^2+(p^2\pm q^2)^2 & =\square_{1,2}\,, \\[0.2cm] (p^2\pm q^2)^2-a^2 & =(r^2\pm s^2)^2. \end{align*}
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
