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Congruent numbers with higher exponents. (English) Zbl 1138.11010

The authors investigate the system of Diophantine equations \[ x^2+ay^m=z^2, \qquad x^2-ay^m=w^2, \] where \(a\) and \(m\) are given positive integers with \(m\geq 2\), and the unknowns (\(x,y,z,w\)) are also positive integers. The numbers \(a\) are called congruent numbers, if the above system of equations is solvable with \(m=2\). They are related to right triangles, whose sides are rational numbers and whose area is an integer.
In this paper it is proved that for \(m\geq 3\) the above system has only finitely many solutions, which means solutions with \(\gcd(x,z)=\gcd(x,w)=\gcd(z,w)=1\), as a consequence of the result of H. Darmon and A. Granville [Bull. Lond. Math. Soc. 27, 513–543 (1995; Zbl 0838.11023)]. Further, they describe a method to determine the primitive solutions, when \(a\) is of the form \(2^u\cdot p^v\), where \(p\) is an odd prime and \(u,v\) are nonnegative integers. As an example the use this algorithm for \(a=3\).

MSC:

11D41 Higher degree equations; Fermat’s equation
11D25 Cubic and quartic Diophantine equations

Citations:

Zbl 0838.11023
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References:

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