## 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

Zbl 0838.11023
Full Text:

### References:

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