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Keywords:
congruent numbers; quadratic equations; higher degree equations
congruent numbers; quadratic equations; higher degree equations
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 $a$.
References:
[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 0349554 | Zbl 0259.10010
[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
[3] Darmon H., Merel L.: ‘Winding quotients and some variants of Fermat’s Last Theorem’. J. reine angew. Math., 490 (1997), 81-100. MR 1468926 | Zbl 0976.11017
[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 2076335 | Zbl 1058.11001
[6] Luca F., Szalay L.: ‘Consecutive binomial coefficients satisfying a quadratic relation’. Publ. Math. Debrecen, to appear. MR 2228483 | Zbl 1121.11025
[7] Ribet K.: ‘On the equation $a^p+2^\alpha b^p+c^p=0$’. Acta Arith., 79 (1997), 7-16. MR 1438112
[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 10.1007/BF01389327 | MR 0700775
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