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Keywords:
exponential diophantine equation; Lucas number; positive divisor
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 $\min (y,z)>1$.
References:
[1] Y. Bilu, G. Hanrot and P. Voutier (with an appendix by M. Mignotte): Existence of primitive divisors of Lucas and Lehmer numbers. J. Reine Angew. Math. 539 (2001), 75–122. MR 1863855
[2] Y. Bugeaud: On some exponential diophantine equations. Monatsh. Math. 132 (2001), 93–97. DOI 10.1007/s006050170046 | MR 1838399 | Zbl 1014.11023
[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
[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
[5] L. Jeśmanowicz: Several remarks on Pythagorean number. Wiadom. Mat. 1 (1955/1956), 196–202. (Polish) MR 0110662
[6] C. Ko: On the diophantine equation $x^{2}=y^{n}+1, xy\ne 0$. Sci.Sin. 14 (1964), 457–460. MR 0183684
[7] M.-H. Le: A note on Jeśmanowicz’ conjecture. Colloq. Math. 64 (1995), 47–51. MR 1341681 | Zbl 0849.11036
[8] L. J. Mordell: Diophantine Equations. Academic Press, London, 1969. MR 0249355 | Zbl 0188.34503
[9] T. Nagell: Sur I’impossibilité de quelques equation á deux indéterminées. Norsk Matem. Forenings Skrifter 13 (1921), 65–82.
[10] N. Terai: The diophantine equation $x^{2}+q^{m}=p^{n}$. Acta Arith. 63 (1993), 351–358. MR 1218462
[11] P. Voutier: Primitive divisors of Lucas and Lehmer sequences. Math. Comp. 64 (1995), 869–888. DOI 10.1090/S0025-5718-1995-1284673-6 | MR 1284673 | Zbl 0832.11009
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