| Title:
|
A note on the diophantine equation $x^2+b^Y=c^z$ (English) |
| Author:
|
Le, Maohua |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
56 |
| Issue:
|
4 |
| Year:
|
2006 |
| Pages:
|
1109-1116 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $a$, $b$, $c$, $r$ be positive integers such that $a^{2}+b^{2}=c^{r}$, $\min (a,b,c,r)>1$, $\gcd (a,b)=1, a$ is even and $r$ is odd. In this paper we prove that if $b\equiv 3\hspace{4.44443pt}(\@mod \; 4)$ and either $b$ or $c$ is an odd prime power, then the equation $x^{2}+b^{y}=c^{z}$ has only the positive integer solution $(x,y,z)=(a,2,r)$ with $\min (y,z)>1$. (English) |
| Keyword:
|
exponential diophantine equation |
| Keyword:
|
Lucas number |
| Keyword:
|
positive divisor |
| MSC:
|
11D61 |
| idZBL:
|
Zbl 1164.11319 |
| idMR:
|
MR2280797 |
| . |
| Date available:
|
2009-09-24T11:41:28Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128133 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
[3] Z.-F. Cao and X.-L. Dong: The diophantine equation $a^{2}+b^{y}=c^{z}$.Proc. Japan Acad. 77A (2001), 1–4. MR 1934716 |
| Reference:
|
[4] Z.-F. Cao, X.-L. Dong and Z. Li: A new conjecture concerning the diophantine equation $x^{2}+b^{y}=c^{z}$.Proc. Japan Acad. 78A (2002), 199–202. MR 1950170 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[8] L. J. Mordell: Diophantine Equations.Academic Press, London, 1969. Zbl 0188.34503, MR 0249355 |
| Reference:
|
[9] T. Nagell: Sur I’impossibilité de quelques equation á deux indéterminées.Norsk Matem. Forenings Skrifter 13 (1921), 65–82. |
| Reference:
|
[10] N. Terai: The diophantine equation $x^{2}+q^{m}=p^{n}$.Acta Arith. 63 (1993), 351–358. MR 1218462, 10.4064/aa-63-4-351-358 |
| Reference:
|
[11] P. Voutier: Primitive divisors of Lucas and Lehmer sequences.Math. Comp. 64 (1995), 869–888. Zbl 0832.11009, MR 1284673, 10.1090/S0025-5718-1995-1284673-6 |
| . |
