Geodesic
Congruent numbers with higher exponents
Communications in Mathematics, Tome 14 (2006) no. 1, pp. 49-55
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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 $a$.
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 $a$.
Classification : 11D09, 11D25, 11D41
Keywords: congruent numbers; quadratic equations; higher degree equations 
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Luca, Florian; Szalay, László. Congruent numbers with higher exponents. Communications in Mathematics, Tome 14 (2006) no. 1, pp. 49-55. http://geodesic.mathdoc.fr/item/COMIM_2006_14_1_a7/

[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. | MR | Zbl

[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

[3] Darmon H., Merel L.: ‘Winding quotients and some variants of Fermat’s Last Theorem’. J. reine angew. Math., 490 (1997), 81-100. | MR | Zbl

[4] Dickson L. E.: History of the theory of numbers. Vol. 2, Diophantine analysis, Washington, 1920, 459-472.

[5] Guy R. K.: Unsolved Problems in Number Theory. (D27, p. 306,) Third Edition, Springer, 2004. | MR | Zbl

[6] Luca F., Szalay L.: ‘Consecutive binomial coefficients satisfying a quadratic relation’. Publ. Math. Debrecen, to appear. | MR | Zbl

[7] Ribet K.: ‘On the equation $a^p+2^\alpha b^p+c^p=0$’. Acta Arith., 79 (1997), 7-16. | MR

[8] Robert S.: ‘Note on a problem of Fibonacci’s’. Proc. London Math. Soc., 11 (1879), 35-44.

[9] Tunnel J. B.: ‘A classical Diophantine’s problem and modular forms of weight $3/2$’. Invent. Math., 72 (1983), 323-334. | DOI | MR