| 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:
|
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| Reference:
|
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| 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:
|
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| 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 |
| . |
