On the number of solutions of $x^2-4m(m+1)y^2=y^2-bz^2=1$
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Abstract:
In this paper, using a result of Ljunggren and some results on primitive prime factors of Lucas sequences of the first kind, we prove the following results by an elementary argument: if $m$ and $b$ are positive integers, then the simultaneous Pell equations \[ x^2-4m(m+1)y^2=y^2-bz^2=1\] possesses at most one solution $(x,y,z)$ in positive integers.References
- W. S. Anglin, Simultaneous Pell equations, Math. Comp. 65 (1996), no. 213, 355–359. MR 1325861, DOI 10.1090/S0025-5718-96-00687-4
- W. S. Anglin, The queen of mathematics, Kluwer Texts in the Mathematical Sciences, vol. 8, Kluwer Academic Publishers Group, Dordrecht, 1995. An introduction to number theory. MR 1334998, DOI 10.1007/978-94-011-0285-8
- A. Baker, Linear forms in the logarithms of algebraic numbers. IV, Mathematika 15 (1968), 204–216. MR 258756, DOI 10.1112/S0025579300002588
- A. Baker and H. Davenport, The equations $3x^{2}-2=y^{2}$ and $8x^{2}-7=z^{2}$, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129–137. MR 248079, DOI 10.1093/qmath/20.1.129
- Michael A. Bennett, On the number of solutions of simultaneous Pell equations, J. Reine Angew. Math. 498 (1998), 173–199. MR 1629862, DOI 10.1515/crll.1998.049
- R. D. Carmichael, On the numerical factors of the arithmetic forms $\alpha ^n \pm \beta ^n$, Ann. Math. (2) 15 (1913), 30-70.
- Ezra Brown, Sets in which $xy+k$ is always a square, Math. Comp. 45 (1985), no. 172, 613–620. MR 804949, DOI 10.1090/S0025-5718-1985-0804949-7
- Charles M. Grinstead, On a method of solving a class of Diophantine equations, Math. Comp. 32 (1978), no. 143, 936–940. MR 491480, DOI 10.1090/S0025-5718-1978-0491480-0
- Kiran S. Kedlaya, Solving constrained Pell equations, Math. Comp. 67 (1998), no. 222, 833–842. MR 1443123, DOI 10.1090/S0025-5718-98-00918-1
- D. H. Lehmer, An extended theory of Lucas’ function, Ann. Math. 31 (1931), 419-449.
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- D. W. Masser and J. H. Rickert, Simultaneous Pell equations, J. Number Theory 61 (1996), no. 1, 52–66. MR 1418319, DOI 10.1006/jnth.1996.0137
- R. G. E. Pinch, Simultaneous Pellian equations, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 1, 35–46. MR 913448, DOI 10.1017/S0305004100064598
- Paulo Ribenboim and Wayne L. McDaniel, The square terms in Lucas sequences, J. Number Theory 58 (1996), no. 1, 104–123. MR 1387729, DOI 10.1006/jnth.1996.0068
- C. L. Siegel, Über einige Anwendungen Diophantischer Approximationen, Abh. Preuss. Akad. Wiss. 1 (1929), 1-70.
- A. Thue, Über Annäherungswerte algebraischer Zahlen, J. reine angew. Math. 135 (1909), 284-305.
- Paul M. Voutier, Primitive divisors of Lucas and Lehmer sequences, Math. Comp. 64 (1995), no. 210, 869–888. MR 1284673, DOI 10.1090/S0025-5718-1995-1284673-6
- Pingzhi Yuan, A note on the divisibility of the generalized Lucas sequences, Fibonacci Quart. 40 (2002), no. 2, 153–156. MR 1902752
Additional Information
- Pingzhi Yuan
- Affiliation: Department of Mathematics, Zhongshan University, Guangzhou 510275, P.R. China
- Email: yuanpz@csru.edu.cn, mcsypz@zsu.edu.cn, yuanpz@mail.csu.edu.cn
- Received by editor(s): September 3, 2002
- Published electronically: January 20, 2004
- Communicated by: David E. Rohrlich
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 1561-1566
- MSC (2000): Primary 11D09; Secondary 11D25
- DOI: https://doi.org/10.1090/S0002-9939-04-07418-0
- MathSciNet review: 2051114