# Diophantine Equations and Congruent Number Equation Solutions

@article{Meskhishvili2015DiophantineEA, title={Diophantine Equations and Congruent Number Equation Solutions}, author={Mamuka Meskhishvili}, journal={arXiv: General Mathematics}, year={2015} }

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… CONTINUE READING

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