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On the diophantine system a 2 + b 2 = c 3 and a x + b y = c z for b is an odd prime

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Abstract

Let a, b and c be fixed coprime positive integers. In this paper we prove that if a 2 + b 2 = c 3 and b is an odd prime, then the equation a x + b y = c z has only the positive integer solution (x, y, z) = (2, 2, 3).

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References

  1. Bilu, Y., Hanrot, G., Voutier, P. M. (with appendix by M. Mignotte): Existence of primitive divisors of Lucas and Lehmer numbers. J. Reine Angew. Math., 539, 75–122 (2001)

    MATH  MathSciNet  Google Scholar 

  2. Cao, Z. F.: A note on the diophantine equation a x + b y = c z. Acta Arith., 91, 85–93 (1999)

    MATH  MathSciNet  Google Scholar 

  3. Darmon, H., Granville, A.: On the equation z m = F(X, Y) and Ax p + By q = Cz r. Bull. London Math. Soc., 27, 513–543 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dong, X. L., Cao, Z. F.: The Terai-Jeśmanowicz conjecture concerning the equation a x +b y = c z. Chinese Math. Ann. 21A, 709–714 (2000) (in Chinese)

    MathSciNet  Google Scholar 

  5. Gel’fond, A. O.: Sur la divisibilité de la dofférence des puissances de deux nombres entieres par une puissance d’un idéal premier. Mat. Sb., 7, 7–25 (1940)

    MathSciNet  Google Scholar 

  6. Hua, L. K., Introduction to number theory, Springer Verlag, Berlin, 1982

    MATH  Google Scholar 

  7. Le, M. H.: Some exponential diophantine equations I: The equation D 1 x 2 − D 2 y 2 = λκ z. J. Number Theory, 55 209–221 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  8. Le, M. H.: A note on the diophantine equation (m 3 − 3m)x + (3m 2 − 1)y = (m 2 +1)z. Proc. Japan Acad. 73A, 148–149 (1997)

    Google Scholar 

  9. Mahler, K.: Zur Approximation algebraischer Zahler I: Über den grössten Primteiler binärer Formen. Math. Ann., 107, 691–730 (1933)

    Article  MATH  MathSciNet  Google Scholar 

  10. Mordell, L. J.: Diophantine equations, Academic Press, London, 1969

    MATH  Google Scholar 

  11. Terai, N.: The diophantine equation a x + b y = c z. Proc. Japan Acad., 70A, 22–26 (1994)

    MathSciNet  Google Scholar 

  12. Voutier, P. M.: Primitive divisors of Lucas and Lehmer sequences. Math. Comp., 64, 869–888 (1995)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Zhanjiang Normal College, Zhanjiang, 524048, P. R. China

    Mao Hua Le

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  1. Mao Hua Le
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Correspondence to Mao Hua Le.

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Supported by the National Natural Science Foundation of China (No. 10271104) and the Guangdong Provincial Natural Science Foundation (No. 04011425)

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Le, M.H. On the diophantine system a 2 + b 2 = c 3 and a x + b y = c z for b is an odd prime. Acta. Math. Sin.-English Ser. 24, 917–924 (2008). https://doi.org/10.1007/s10114-007-6140-x

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  • DOI: https://doi.org/10.1007/s10114-007-6140-x

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