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Title: Congruent numbers with higher exponents (English)
Author: Luca, Florian
Author: Szalay, László
Language: English
Journal: Acta Mathematica Universitatis Ostraviensis
ISSN: 1214-8148
Volume: 14
Issue: 1
Year: 2006
Pages: 49-55
Category: math
Summary: 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 $a$. (English)
Keyword: congruent numbers
Keyword: quadratic equations
Keyword: higher degree equations
MSC: 11D09
MSC: 11D25
MSC: 11D41
idZBL: Zbl 1138.11010
idMR: MR2298913
Date available: 2009-12-29T09:20:08Z
Last updated: 2013-10-22
Stable URL:
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