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THE EXPONENTIAL DIOPHANTINE EQUATION nx + (n + 1)y = (n + 2)z REVISITED

Published online by Cambridge University Press:  01 September 2009

BO HE  and
ALAIN TOGBÉ
BO HE
Affiliation:
College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610068, People's Republic of China e-mail: [email protected]
ALAIN TOGBÉ
Affiliation:
Mathematics Department, Purdue University North Central, 1401 S, U.S. 421, Westville, IN 46391, USA e-mail: [email protected]
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Abstract

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Let n be a positive integer. In this paper, we consider the diophantine equation We prove that this equation has only the positive integer solutions (n, x, y, z) = (1, t, 1, 1), (1, t, 3, 2), (3, 2, 2, 2). Therefore we extend the work done by Leszczyński (Wiadom. Mat., vol. 3, 1959, pp. 37–39) and Makowski (Wiadom. Mat., vol. 9, 1967, pp. 221–224).

Keywords

Primary 11D61secondary 11B3911J86
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 3 , September 2009 , pp. 659 - 667
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Abouzaid, M., Les nombres de Lucas et Lehmer sans diviseur primitif, J. Théor. Nombres Bordeaux 18 (2006), 299–313.CrossRefGoogle Scholar
2.Bang, A. S., Tialtheoretiske Undersøgelser, Tidsskrift Mat. 4 (1886), 70–80, 130–137.Google Scholar
3.Bilu, Y., Hanrot, G. and Voutier, P. (with an appendix by M. Mignotte), Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math. 539 (2001), 75–122.Google Scholar
4.Birkhoff, G. D. and Vandiver, H. S., On the integral divisors of an − bn, Ann. Math. 5 (2) (1904), 173–180.CrossRefGoogle Scholar
5.Cao, Z. F. and Dong, X., On Terai's conjecture, Proc. Japan Acad. A Math. Sci. 74 (1998), 127–129.Google Scholar
6.Cao, Z. F., A note on the the Diophantine equation ax + by = cz, Acta Arth. 91 (1999), 85–93.CrossRefGoogle Scholar
7.Cao, Z. F., Dong, X. and Li, Z., A new conjecture concerning the Diophantine equation ax + by = cz, Proc. Japan Acad. A Math. Sci. 78 (2002), 199–202.Google Scholar
8.Ivorra, W., Sur les équations xp + 2βyp = z 2 et xp + 2βyp = 2z 2, Acta Arith. 108 (2003), 327–338.CrossRefGoogle Scholar
9.Jeśmanowicz, L., Some remarks on Pythagorean numbers, Wiadom. Mat., 1 (1956), 196–202.Google Scholar
10.Khinchin, A. Y., Continued fractions, 3rd edition (P. Noordhof, Groningen, The Netherlands, 1963).Google Scholar
11.Laurent, M., Mignotte, M. and Nesterenko, Yu., Formes linéaires en deux logarithmes et déterminants d'interpolation, J. Number Theory 55 (1995), 285–321.CrossRefGoogle Scholar
12.Leszczyński, B., On the equation nx + (n + 1)y = (n + 2)z, Wiadom. Mat. 3 (1959), 37–39.Google Scholar
13.LeVeque, W. J., On the equation ax − by = 1, Am. J. Math. 74 (1952), 325–331.CrossRefGoogle Scholar
14.Makowski, A., On the equation nx + (n + 1)y = (n + 2)z, Wiadom. Mat. 9 (1967), 221–224.Google Scholar
15.Sierpiński, W., On the equation 3x + 4y = 5z, Wiadom. Mat. 1 (1956), 194–195.Google Scholar
16.Siksek, S., On the Diophantine equation x 2 = yp + 2k zp, J Théor. Nombres Bordeaux 15 (2003), 839–846.CrossRefGoogle Scholar
17.Terai, N., The Diophantine equation x 2 + qm = pn, Acta Arith. 63 (1993), 351–358.CrossRefGoogle Scholar
18.Terai, N., The Diophantine equation ax + by = cz, Proc. Japan Acad. A Math. Sci. 70 (1994), 22–26.Google Scholar
19.Terai, N., The Diophantine equation ax + by = cz II, Proc. Japan Acad. A Math. Sci. 71 (1995), 109–110.Google Scholar
20.Terai, N., The Diophantine equation ax + by = cz III, Proc. Japan Acad. A Math. Sci. 72 (1996), 20–22.Google Scholar
21.Terai, N. and Tawakuwa, K., A note on the the Diophantine equation ax + by = cz, Proc. Japan Acad. A Math. Sci. 73 (1997), 161–164.Google Scholar
22.Zsigmondy, K., Zur Theorie der Potenzreste, Monatsh. Math. 3 (1892), 265–284.CrossRefGoogle Scholar