# Article

 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: http://hdl.handle.net/10338.dmlcz/137483 . Reference: [1] Alter R., Curtz T. B., Kubota K. K: ‘Remarks and results on congruent numbers’., Proc. 3rd S. E. Conf. Combin. Graph Theory Comput., Congr. Num., 6 (1972), 27-35. Zbl 0259.10010, MR 0349554 Reference: [2] Darmon H., Granville A.: ‘On the equations $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$’., Bull. London Math. Soc., 27 (1995), 513–543. MR 1348707 Reference: [3] Darmon H., Merel L.: ‘Winding quotients and some variants of Fermat’s Last Theorem’., J. reine angew. Math., 490 (1997), 81-100. Zbl 0976.11017, MR 1468926 Reference: [4] Dickson L. E.: History of the theory of numbers., Vol. 2, Diophantine analysis, Washington, 1920, 459-472. Reference: [5] Guy R. K.: Unsolved Problems in Number Theory., (D27, p. 306,) Third Edition, Springer, 2004. Zbl 1058.11001, MR 2076335 Reference: [6] Luca F., Szalay L.: ‘Consecutive binomial coefficients satisfying a quadratic relation’., Publ. Math. Debrecen, to appear. Zbl 1121.11025, MR 2228483 Reference: [7] Ribet K.: ‘On the equation $a^p+2^\alpha b^p+c^p=0$’., Acta Arith., 79 (1997), 7-16. MR 1438112 Reference: [8] Robert S.: ‘Note on a problem of Fibonacci’s’., Proc. London Math. Soc., 11 (1879), 35-44. Reference: [9] Tunnel J. B.: ‘A classical Diophantine’s problem and modular forms of weight $3/2$’., Invent. Math., 72 (1983), 323-334. MR 0700775, 10.1007/BF01389327 .

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