Glasnik Matematicki, Vol. 57, No. 2 (2022), 185-201. \( \)

ON THE MULTIPLICITY IN PILLAI'S PROBLEM WITH FIBONACCI NUMBERS AND POWERS OF A FIXED PRIME

Herbert Batte, Mahadi Ddamulira, Juma Kasozi and Florian Luca

Department of Mathematics, Makerere University, Kampala, Uganda
e-mail:hbatte91@gmail.com

Department of Mathematics, Makerere University, Kampala, Uganda
e-mail:mahadi.ddamulira@mak.ac.ug

Department of Mathematics, Makerere University, Kampala, Uganda
e-mail:juma.kasozi@mak.ac.ug

School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa, &, Research Group in Algebraic Structures and Applications, King Abdulaziz University, Jeddah, Saudi Arabia, &, Max Planck Institute for Software Systems, Saarbrücken, Germany, &, Centro de Ciencias Matemáticas UNAM, Morelia, Mexico
e-mail:Florian.Luca@wits.ac.za

Abstract.   Let \( \{F_n\}_{n\geq 0} \) be the sequence of Fibonacci numbers and let \(p\) be a prime. For an integer \(c\) we write \(m_{F,p}(c)\) for the number of distinct representations of \(c\) as \(F_k-p^\ell\) with \(k\ge 2\) and \(\ell\ge 0\). We prove that \(m_{F,p}(c)\le 4\).

2020 Mathematics Subject Classification.   11B39, 11D61, 11J86

Key words and phrases.   Fibonacci numbers, prime numbers, linear forms in logarithms, Pillai's problem

Full text (PDF) (access from subscribing institutions only)

https://doi.org/10.3336/gm.57.2.02

References:

  1. F. Amoroso and E. Viada, Small points on subvarieties of a torus, Duke Math. J. 150 (2009), 407–442.
    MathSciNet    CrossRef

  2. A. Baker and G. Wüstholz, Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), 19–62.
    MathSciNet    CrossRef

  3. J. J. Bravo, F. Luca and K. Yazán, On Pillai's problem with Tribonacci numbers and Powers of 2, Bull. Korean Math. Soc. 54 (2017), 1069–1080.
    MathSciNet    CrossRef

  4. Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers, Ann. of Math. (2) 163 (2006), 969–1018.
    MathSciNet    CrossRef

  5. K. C. Chim, I. Pink and V. Ziegler, On a variant of Pillai's problem II, J. Number Theory 183 (2018), 269–290.
    MathSciNet    CrossRef

  6. M. Ddamulira, F. Luca and M. Rakotomalala, On a problem of Pillai with Fibonacci numbers and powers of 2, Proc. Indian Acad. Sci. Math. Sci. 127 (2017), 411–421.
    MathSciNet    CrossRef

  7. M. Ddamulira, On a problem of Pillai with Fibonacci numbers and powers of 3. Bol. Soc. Mat. Mex. 26 (2020), 263–277.
    MathSciNet    CrossRef

  8. S. Gúzman and F. Luca, Linear combinations of factorials and \(s\)-units in a binary recurrence sequence, Ann. Math. Qué. 38 (2014), 169–188.
    MathSciNet    CrossRef

  9. M. Laurent, M. Mignotte and Y. Nesterenko, Formes linéaires en deux logarithmes et déterminants d'interpolation, J. Number Theory 55 (1995), 285–321.
    MathSciNet    CrossRef

  10. E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II, (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 125–180; translation in Izv. Math. 64 (2000), 1217–1269.
    MathSciNet    CrossRef

  11. R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp. 76 (2007), 2087–2094.
    MathSciNet    CrossRef

  12. M. R. Murty and J. Esmonde, Problems in algebraic number theory, Second edition. Graduate Texts in Mathematics, 190, Springer Verlag, New York, 2005.
    MathSciNet    CrossRef

  13. S. S. Pillai, On \(a^x - b^y = c\), J. Indian Math. Soc. N. Ser. 2 (1936), 119–122.
Glasnik Matematicki Home Page