Previous |  Up |  Next

Article

Keywords:
exponential diophantine equation; upper bound for solutions; singular number
Summary:
For any positive integer $D$ which is not a square, let $(u_1,v_1)$ be the least positive integer solution of the Pell equation $u^2-Dv^2=1,$ and let $h(4D)$ denote the class number of binary quadratic primitive forms of discriminant $4D$. If $D$ satisfies $2\nmid D$ and $v_1h(4D)\equiv 0 \pmod D$, then $D$ is called a singular number. In this paper, we prove that if $(x,y,z)$ is a positive integer solution of the equation $x^y+y^x=z^z$ with $2\mid z$, then maximum $\max \{x,y,z\}<480000$ and both $x$, $y$ are singular numbers. Thus, one can possibly prove that the equation has no positive integer solutions $(x,y,z)$.
References:
[1] Bilu, Y., Hanrot, G., Voutier, P. M.: Existence of primitive divisors of Lucas and Lehmer numbers. J. Reine Angew. Math. 539 (2001), 75-122. DOI 10.1515/crll.2001.080 | MR 1863855 | Zbl 0995.11010
[2] Birkhoff, G. D., Vandiver, H. S.: On the integral divisors of $a^n-b^n$. Ann. of Math. (2) 5 (1904), 173-180. DOI 10.2307/2007263 | MR 1503541 | Zbl 35.0205.01
[3] Buell, D. A.: Computer computation of class groups of quadratic number fields. Congr. Numerantium 22 Conf. Proc. Numerical Mathematics and Computing, Winnipeg 1978 (1979), 3-12 McCarthy et al. MR 0541910 | Zbl 0424.12001
[4] 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. DOI 10.1017/S0305004199003692 | MR 1713116 | Zbl 0940.11019
[5] Deng, Y., Zhang, W.: On the odd prime solutions of the Diophantine equation $x^y+y^x=z^z$. Abstr. Appl. Anal. 2014 (2014), Art. ID 186416, 4 pages. DOI 10.1155/2014/186416 | MR 3240527
[6] Le, M.: Some exponential Diophantine equations. I: The equation $D_1x^2-D_2y^2=\lambda k^z$. J. Number Theory 55 (1995), 209-221. DOI 10.1006/jnth.1995.1138 | MR 1366571 | Zbl 0852.11015
[7] Le, M.: On the Diophantine equation $y^x-x^y=z^2$. Rocky Mt. J. Math. (2007), 37 1181-1185. DOI 10.1216/rmjm/1187453105 | MR 2360292 | Zbl 1146.11019
[8] Liu, Y. N., Guo, X. Y.: A Diophantine equation and its integer solutions. Acta Math. Sin., Chin. Ser. 53 (2010), 853-856. MR 2722920 | Zbl 1240.11066
[9] Luca, F., Mignotte, M.: On the equation $y^x\pm x^y=z^2$. Rocky Mt. J. Math. 30 (2000), 651-661. DOI 10.1216/rmjm/1022009287 | MR 1787004 | Zbl 1014.11024
[10] Mollin, R. A., Williams, H. C.: Computation of the class number of a real quadratic field. Util. Math. 41 (1992), 259-308. MR 1162532 | Zbl 0757.11036
[11] Mordell, L. J.: Diophantine Equations. Pure and Applied Mathematics 30, Academic Press, London (1969). MR 0249355 | Zbl 0188.34503
[12] 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), 70 1311-1328 corrig. ibid. 72 521-523 2003. DOI 10.1090/S0025-5718-00-01234-5 | MR 1933835 | Zbl 0987.11065
[13] Wu, H.: The application of BHV theorem to the Diophantine equation $x^y+y^x=z^z$. Acta Math. Sin., Chin. Ser. 58 (2015), 679-684. MR 3443204 | Zbl 06610974
[14] Zhang, Z., Luo, J., Yuan, P.: On the Diophantine equation $x^y-y^x=c^z$. Colloq. Math. 128 (2012), 277-285. DOI 10.4064/cm128-2-13 | MR 3002356 | Zbl 1297.11017
[15] Zhang, Z., Luo, J., Yuan, P.: On the Diophantine equation $x^y+y^x=z^z$. Chin. Ann. Math., Ser. A (2013), 34A 279-284. MR 3114411 | Zbl 1299.11037
[16] Zhang, Z., Yuan, P.: On the Diophantine equation $ax^y+by^z+cz^x=0$. Int. J. Number Theory 8 (2012), 813-821. DOI 10.1142/S1793042112500467 | MR 2904932 | Zbl 1271.11040
Partner of
EuDML logo