Congruent numbers with higher exponents

Florian Luca; László Szalay

Acta Mathematica Universitatis Ostraviensis (2006)

  • Volume: 14, Issue: 1, page 49-55
  • ISSN: 1804-1388

Abstract

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This paper investigates the system of equations x 2 + a y m = z 1 2 , x 2 - a y m = z 2 2 in positive integers x , y , z 1 , z 2 , where a and m are positive integers with m ≥ 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 .

How to cite

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Luca, Florian, and Szalay, László. "Congruent numbers with higher exponents." Acta Mathematica Universitatis Ostraviensis 14.1 (2006): 49-55. <http://eudml.org/doc/35162>.

@article{Luca2006,
abstract = {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 $a$.},
author = {Luca, Florian, Szalay, László},
journal = {Acta Mathematica Universitatis Ostraviensis},
keywords = {congruent numbers; quadratic equations; higher degree equations; congruent numbers; Fermat type equations; primitive solutions},
language = {eng},
number = {1},
pages = {49-55},
publisher = {University of Ostrava},
title = {Congruent numbers with higher exponents},
url = {http://eudml.org/doc/35162},
volume = {14},
year = {2006},
}

TY - JOUR
AU - Luca, Florian
AU - Szalay, László
TI - Congruent numbers with higher exponents
JO - Acta Mathematica Universitatis Ostraviensis
PY - 2006
PB - University of Ostrava
VL - 14
IS - 1
SP - 49
EP - 55
AB - 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 $a$.
LA - eng
KW - congruent numbers; quadratic equations; higher degree equations; congruent numbers; Fermat type equations; primitive solutions
UR - http://eudml.org/doc/35162
ER -

References

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  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. (1972) Zbl0259.10010MR0349554
  2. Darmon H., Granville A., ‘On the equations z m = F ( x , y ) and A x p + B y q = C z r ’, , Bull. London Math. Soc., 27 (1995), 513–543. (1995) MR1348707
  3. Darmon H., Merel L., ‘Winding quotients and some variants of Fermat’s Last Theorem’, , J. reine angew. Math., 490 (1997), 81-100. (1997) Zbl0976.11017MR1468926
  4. Dickson L. E., History of the theory of numbers, , Vol. 2, Diophantine analysis, Washington, 1920, 459-472. (1920) 
  5. Guy R. K., Unsolved Problems in Number Theory, , (D27, p. 306,) Third Edition, Springer, 2004. Zbl1058.11001MR2076335
  6. Luca F., Szalay L., ‘Consecutive binomial coefficients satisfying a quadratic relation’, , Publ. Math. Debrecen, to appear. Zbl1121.11025MR2228483
  7. Ribet K., ‘On the equation a p + 2 α b p + c p = 0 ’, , Acta Arith., 79 (1997), 7-16. (1997) MR1438112
  8. Robert S., ‘Note on a problem of Fibonacci’s’, , Proc. London Math. Soc., 11 (1879), 35-44. 
  9. Tunnel J. B., 10.1007/BF01389327, , Invent. Math., 72 (1983), 323-334. (1983) MR0700775DOI10.1007/BF01389327

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