Complete solution of the Diophantine equation x y + y x = z z

Mihai Cipu

Czechoslovak Mathematical Journal (2019)

  • Volume: 69, Issue: 2, page 479-484
  • ISSN: 0011-4642

Abstract

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The triples ( x , y , z ) = ( 1 , z z - 1 , z ) , ( x , y , z ) = ( z z - 1 , 1 , z ) , where z ∈ 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.

How to cite

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Cipu, Mihai. "Complete solution of the Diophantine equation $x^y+y^x=z^z$." Czechoslovak Mathematical Journal 69.2 (2019): 479-484. <http://eudml.org/doc/294782>.

@article{Cipu2019,
abstract = {The triples $(x,y,z)=(1,z^z-1,z)$, $(x,y,z)=(z^z-1,1,z)$, where $z\in \mathbb \{N\}$, satisfy the equation $x^y+y^x=z^z$. In this paper it is shown that the same equation has no integer solution with $\min \lbrace x,y,z\rbrace > 1$, thus a conjecture put forward by Z. Zhang, J. Luo, P. Z. Yuan (2013) is confirmed.},
author = {Cipu, Mihai},
journal = {Czechoslovak Mathematical Journal},
keywords = {exponential Diophantine equation; sieving; modular computations},
language = {eng},
number = {2},
pages = {479-484},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Complete solution of the Diophantine equation $x^y+y^x=z^z$},
url = {http://eudml.org/doc/294782},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Cipu, Mihai
TI - Complete solution of the Diophantine equation $x^y+y^x=z^z$
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 2
SP - 479
EP - 484
AB - The triples $(x,y,z)=(1,z^z-1,z)$, $(x,y,z)=(z^z-1,1,z)$, where $z\in \mathbb {N}$, satisfy the equation $x^y+y^x=z^z$. In this paper it is shown that the same equation has no integer solution with $\min \lbrace x,y,z\rbrace > 1$, thus a conjecture put forward by Z. Zhang, J. Luo, P. Z. Yuan (2013) is confirmed.
LA - eng
KW - exponential Diophantine equation; sieving; modular computations
UR - http://eudml.org/doc/294782
ER -

References

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  1. Bugeaud, Y., 10.1017/S0305004199003692, Math. Proc. Camb. Philos. Soc. 127 (1999), 373-381. (1999) Zbl0940.11019MR1713116DOI10.1017/S0305004199003692
  2. Deng, Y., Zhang, W., 10.1155/2014/186416, Abstr. Appl. Anal. 2014 (2014), Article ID 186416, 4 pages. (2014) MR3240527DOI10.1155/2014/186416
  3. Du, X., 10.21136/CMJ.2017.0645-15, Czech. Math. J. 67 (2017), 645-653. (2017) Zbl06770122MR3697908DOI10.21136/CMJ.2017.0645-15
  4. Mollin, R. A., Williams, H. C., Computation of the class number of a real quadratic field, Util. Math. 41 (1992), 259-308. (1992) Zbl0757.11036MR1162532
  5. The PARI-Group: PARI/GP version 2.3.5, Univ. Bordeaux (2010), Available at http://pari.math.u-bordeaux.fr/download.html. (2010) 
  6. Poorten, A. J. van der, Riele, H. J. J. te, Williams, H. C., 10.1090/S0025-5718-00-01234-5, Math. Comput. 70 (2001), 1311-1328 corrigenda and addition ibid. 72 521-523 2003. (2001) Zbl0987.11065MR1933835DOI10.1090/S0025-5718-00-01234-5
  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. (2015) Zbl1349.11077MR3443204
  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. (2013) Zbl1299.11037MR3114411

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