| Title:
|
Complete solution of the Diophantine equation $x^y+y^x=z^z$ (English) |
| Author:
|
Cipu, Mihai |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
2 |
| Year:
|
2019 |
| Pages:
|
479-484 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The triples $(x,y,z)=(1,z^z-1,z)$, $(x,y,z)=(z^z-1,1,z)$, where $z\in \Bbb N$, satisfy the equation $x^y+y^x=z^z$. In this paper it is shown that the same equation has no integer solution with $\min \{x,y,z\} > 1$, thus a conjecture put forward by Z. Zhang, J. Luo, P. Z. Yuan (2013) is confirmed. (English) |
| Keyword:
|
exponential Diophantine equation |
| Keyword:
|
sieving |
| Keyword:
|
modular computations |
| MSC:
|
11A15 |
| MSC:
|
11D61 |
| idZBL:
|
Zbl 07088800 |
| idMR:
|
MR3959960 |
| DOI:
|
10.21136/CMJ.2018.0395-17 |
| . |
| Date available:
|
2019-05-24T09:00:44Z |
| Last updated:
|
2021-07-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147740 |
| . |
| Reference:
|
[1] Bugeaud, Y.: Linear forms in $p$-adic logarithms and the Diophantine equation $(x^n-1)/(x-1)=y^q$.Math. Proc. Camb. Philos. Soc. 127 (1999), 373-381. Zbl 0940.11019, MR 1713116, 10.1017/S0305004199003692 |
| Reference:
|
[2] Deng, Y., Zhang, W.: On the odd prime solutions of the Diophantine equation $x^y+y^x=z^z$.Abstr. Appl. Anal. 2014 (2014), Article ID 186416, 4 pages. MR 3240527, 10.1155/2014/186416 |
| Reference:
|
[3] Du, X.: On the exponential Diophantine equation $x^y+y^x=z^z$.Czech. Math. J. 67 (2017), 645-653. Zbl 06770122, MR 3697908, 10.21136/CMJ.2017.0645-15 |
| Reference:
|
[4] Mollin, R. A., Williams, H. C.: Computation of the class number of a real quadratic field.Util. Math. 41 (1992), 259-308. Zbl 0757.11036, MR 1162532 |
| Reference:
|
[5] : The PARI-Group: PARI/GP version 2.3.5.Univ. Bordeaux (2010), Available at http://pari.math.u-bordeaux.fr/download.html\kern0pt. |
| Reference:
|
[6] Poorten, A. J. van der, Riele, H. J. J. te, Williams, H. C.: Computer verification of the Ankeny-Artin-Chowla Conjecture for all primes less than 100000000000.Math. Comput. 70 (2001), 1311-1328 corrigenda and addition ibid. 72 521-523 2003. Zbl 0987.11065, MR 1933835, 10.1090/S0025-5718-00-01234-5 |
| Reference:
|
[7] Wu, H. M.: The application of the BHV theorem to the Diophantine equation $x^y+y^x=z^z$.Acta Math. Sin., Chin. Ser. 58 (2015), Chinese 679-684. Zbl 1349.11077, MR 3443204 |
| Reference:
|
[8] Zhang, Z., Luo, J., Yuan, P.: On the Diophantine equation $x^y-y^x=z^z$.Chin. Ann. Math., Ser. A 34 (2013), 279-284 Chinese. Zbl 1299.11037, MR 3114411 |
| . |
