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congruent numbers; quadratic equations; higher degree equations
This paper investigates the system of equations \[x^2+ay^m=z_1^2, \quad \quad x^2-ay^m=z_2^2\] in positive integers $x$, $y$, $z_1$, $z_2$, where $a$ and $m$ are positive integers with $m\ge 3$. In case of $m=2$ we would obtain the classical problem of congruent numbers. We provide a procedure to solve the simultaneous equations above for a class of the coefficient $a$ with the condition $\gcd (x,z_1)=\gcd (x,z_2)=\gcd (z_1,z_2)=1$. Further, under same condition, we even prove a finiteness theorem for arbitrary nonzero $a$.
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