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On members of Lucas sequences which are products of factorials

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Abstract

We show that if \(\{U_n\}_{n\ge 0}\) is a Lucas sequence, then the largest n such that \(|U_n|=m_1!m_2!\cdots m_k!\) with \(1\le m_1\le m_2\le \cdots \le m_k\) satisfies \(n<\) 62,000. When the roots of the Lucas sequence are real, we have \(n\in \{1, 2, 3, 4, 6, 12\}\). As a consequence, we show that if \(\{X_n\}_{n\ge 1}\) is the sequence of X-coordinates of a Pell equation \(X^2-dY^2=\pm 1\) with a non-zero integer \(d>1\), then \(X_n=m!\) implies \(n=1\).

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References

  1. Bennett, M.A., Martin, G., O’Bryant, K., Rechnitzer, A.: Explicit bounds for primes in arithmetic progressions. Ill. J. Math. 62(1–4), 427–532 (2018)

    Article  MathSciNet  Google Scholar 

  2. Bérczes, A., Dujella, A., Hajdu, L., Saradha, N., Tijdeman, R.: Products of factorials which are powers. Acta Arith. 190, 339–350 (2019)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. de Koninck, J.-M., Doyon, N., Razafindrasoanaivolala, A.A.B., Verreault, W.: Bounds for the counting function of Jordan–Pólya numbers. Archivum Math. (to appear)

  5. Ljunggren, W.: Über die Gleichung \(x^4-Dy^2=1\). Arch. Math. Naturvid. 45, 61–70 (1942)

    MathSciNet  MATH  Google Scholar 

  6. Luca, F.: Products of factorials in binary recurrence sequences. Rocky Mt. J. Math. 29, 1387–1411 (1999)

    Article  MathSciNet  Google Scholar 

  7. Luca, F., Stănică, P.: \(F_1F_2F_3F_4F_5F_6F_8F_{10}F_{12}=11!\). Portugaliae Math. 63, 251–260 (2006)

    MathSciNet  MATH  Google Scholar 

  8. Luca, F.: Fibonacci numbers with the Lehmer property. Bull. Pol. Acad. Sci. Math. 55, 7–15 (2007)

    Article  MathSciNet  Google Scholar 

  9. Montgomery, H.L., Vaughan, R.C.: The large sieve. Mathematika 20, 119–134 (1973)

    Article  MathSciNet  Google Scholar 

  10. Sun, Q., Yuan, P.Z.: A note on the Diophantine equation \(x^4-Dy^2=1\). Sichuan Daxue Xuebao 34, 265–268 (1997)

    MathSciNet  Google Scholar 

  11. Voutier, P.: Primitive divisors of Lucas and Lehmer sequences, III. Math. Proc. Camb. Philos. Soc. 123, 407–419 (1998)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We would like to thank the referee for carefully reading the manuscript and for pointing out the small computational error as well as the list of missing solutions and other useful suggestions in the paper. Part of this work was done when the first author visited the School of Maths of Wits University in December 2018. He thanks this Institution for hospitality and CoEMaSS Grant RTNUM18 and National Research Foundation (Grant No. CPRR160325161141) for financial support. Part of this work was done when the second author visited the Max Planck Institute for Mathematics in Bonn, Germany in Fall of 2019. This author thanks MPIM for hospitality.

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Authors and Affiliations

  1. Stat-Math Unit, Indian Statistical Institute, 7, S. J. S. Sansanwal Marg, New Delhi, 110016, India

    Shanta Laishram

  2. School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits, 2050, South Africa

    Florian Luca

  3. Research Group in Algebraic Structures and Applications, King Abdulaziz University, Jeddah, Saudi Arabia

    Florian Luca

  4. Centro de Ciencias Matemáticas, UNAM, Morelia, Mexico

    Florian Luca

  5. Department of Pure and Applied Mathematics, University of Johannesburg, PO Box 524, Auckland Park, 2006, South Africa

    Mark Sias

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  1. Shanta Laishram
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  2. Florian Luca
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  3. Mark Sias
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Correspondence to Shanta Laishram.

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Communicated by Adrian Constantin.

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Laishram, S., Luca, F. & Sias, M. On members of Lucas sequences which are products of factorials. Monatsh Math 193, 329–359 (2020). https://doi.org/10.1007/s00605-020-01455-y

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  • DOI: https://doi.org/10.1007/s00605-020-01455-y

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