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Linear combinations of prime powers in X-coordinates of Pell equations

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Abstract

Let \(\{X_\ell \}_{\ell \ge 1}\) be the sequence of X-coordinates of the positive integer solutions (X, Y) of the Pell equation \(X^2-dY^2=\pm 1\) corresponding to a nonsquare integer \(d>1\). We show that there are only a finite number of nonsquare integers \(d > 1\) such that there are at least two different elements of the sequence \(\{X_\ell \}_{\ell \ge 1}\) that can be represented as a linear combination of prime powers with fixed primes and coefficients, restricted to the condition that the exponent of the largest prime is the greatest of all exponents. Moreover, we solve explicitly the case in which two of the X-coordinates above are a sum of a power of two and a power of three under the above condition on the exponents. This work is motivated by the recent paper Bertók et al. (Int J Number Theory 13(02):261–271, 2017).

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Notes

  1. Zeckendorf’s theorem claims that every positive integer N has a unique representation as sum of non-consecutive Fibonacci numbers. That is, \( N = F_{k_1}+\cdots +F_{k_r},\) where \(k_{i+1}-k_{i} \ge 2\) for all \(i=1,2,\ldots ,r-1\).

  2. For an integer \(k\ge 2\), the k-generalized Fibonacci sequence \(F^{(k)}\) satisfies that its first k terms are \(0, \ldots , 0, 1\) and each term afterwards is the sum of the preceding k terms. For \(k = 2\), this reduces to the familiar Fibonacci numbers, while for \(k=3\) these are the Tribonacci numbers.

  3. A repdigit in base b or b-repdigit is an integer whose digits in its base b-representation are all equal.

References

  1. Baker, A., Davenport, H.: The equations \(3x^2-2 =y^2\) and \(8x^2-7=z^2\). Q. J. Math. Oxf. Ser. (2) 20, 129–137 (1969)

    Article  Google Scholar 

  2. Bertók, C., Hajdu, L., Pink, I., Rábai, Z.: Linear combinations of prime powers in binary recurrence sequences. Int. J. Number Theory 13(02), 261–271 (2017)

    Article  MathSciNet  Google Scholar 

  3. Bir, K., Luca, F., Togbé, A.: On the \(x\)-coordinates of Pell equations which are Fibonacci numbers. Colloq. Math. 149, 75–85 (2018)

    MathSciNet  MATH  Google Scholar 

  4. Bravo, E.F., Gómez, C.A., Luca, F.: \(x\)-Coordinates of Pell equations as sums of two tribonacci numbers. Period. Math. Hung. 77, 175–190 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Cohen, H.: Number Theory. Volume II: Analytic and Modern Tools. Graduate Texts in Mathematics. Springer, New York (2007)

    MATH  Google Scholar 

  7. Ddamulira, M., Luca, F.: On the \(x\)-coordinates of Pell equations which are \(k\)-generalized Fibonacci numbers. J. Number Theory (2018). https://doi.org/10.1016/j.jnt.2019.07.006

  8. Díaz Alvarado, S., Luca, F.: Fibonacci numbers which are sums of two repdigits. In: Proceedings of the XIVth International Conference on Fibonacci Numbers and Their Applications, Sociedad Matemática Mexicana, Aportaciones Matemáticas, Investigación, vol. 20, pp. 97–108 (2011)

  9. Dossavi-Yovo, A., Luca, F., Togbé, A.: On the \(x\)-coordinates of Pell equations which are rep-digits. Publ. Math. Debr. 88, 381–399 (2016)

    Article  MathSciNet  Google Scholar 

  10. Dujella, A., Pethő, A.: A generalization of a theorem of Baker and Davenport. Q. J. Math. Oxf. 49, 291–306 (1998)

    Article  MathSciNet  Google Scholar 

  11. Faye, B., Luca, F.: On \(x\)-coordinates of Pell equations which are repdigits. Fibonacci Q. 56, 52–62 (2018)

    MATH  Google Scholar 

  12. Gómez, C.A., Luca, F.: Zeckendorf representations with at most two terms to \(x\)-coordinates of Pell equations. Sci. China Math. (2019). https://doi.org/10.1007/s11425-017-9283-6

  13. Luca, F.: Distinct digits in base \(b\) expansions of linear recurrence sequences. Quaest. Math. 23, 389–404 (2000)

    Article  MathSciNet  Google Scholar 

  14. Luca, F., Stănică, P.: Fibonacci numbers of the form \(p^a \pm p^b\). Congr. Numer. 194, 177–183 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Luca, F., Szalay, L.: Fibonacci numbers of the form \(p^a \pm p^b + 1\). Fibonacci Q. 45, 98–103 (2007)

    MATH  Google Scholar 

  16. Luca, F., Togbé, A.: On the \(x\)-coordinates of Pell equations which are Fibonacci numbers. Math. Scand. 122, 18–30 (2018)

    Article  MathSciNet  Google Scholar 

  17. Luca, F., Montejano, A., Szalay, L., Togbé, A.: On the \(x\)-coordinates of Pell equations which are Tribonacci numbers. Acta Arith. (2019)

  18. Marques, D., Togbé, A.: Fibonacci and Lucas numbers of the form \(2^a + 3^b + 5^c\). Proc. Jpn. Acad. Ser. A 89, 47–50 (2013)

    Article  Google Scholar 

  19. Matveev, E.M.: An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. Izv. Math. 64, 1217–1269 (2000)

    Article  MathSciNet  Google Scholar 

  20. Meher, N.K., Rout, S.S.: Linear combinations of prime powers in sums of terms of binary recurrence sequences. Lith. Math. J. 57(4), 506–520 (2017). https://doi.org/10.1007/s10986-017-9374-z

    Article  MathSciNet  MATH  Google Scholar 

  21. Pethő, A.: Perfect powers in second order linear recurrences. J. Number Theory 15, 5–13 (1982)

    Article  MathSciNet  Google Scholar 

  22. Pethő, A.: The Pell sequence contains only trivial perfect powers. Proceedings of Sets Graphs and Numbers. Coll. Math. Soc. János Bolyai 60, 561–568 (1992)

    MATH  Google Scholar 

  23. Pethő, A., Tichy, R.F.: \(S\)-unit equations, linear recurrences and digit expansions. Publ. Math. Debr. 42, 145–154 (1993)

    MathSciNet  MATH  Google Scholar 

  24. Shorey, T.N., Stewart, C.L.: On the Diophantine equation \(ax^{2t} + bx^{ty} +cy^2 = d\) and pure powers in recurrence sequences. Math. Scand. 52, 24–36 (1983)

    Article  MathSciNet  Google Scholar 

  25. Stewart, C.L.: On the representation of an integer in two different bases. J. Reine Angew. Math. 319, 63–72 (1980)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank the referee for a careful reading of the manuscript and for several suggestions which improved the presentation of our paper. This paper started during the ALTENCOA 8 conference in Universidad del Cauca, Popayan, Colombia, where all authors attended and presented talks. The authors thank the organizers of that conference for a great working environment.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad del Valle, Calle 13 No 100–00, Cali, Colombia

    Harold Erazo & Carlos A. Gómez

  2. School of Mathematics, University of the Witwatersrand, Private Bag X3, Wits, 2050, South Africa

    Florian Luca

  3. Department of Mathematics, Faculty of Sciences, University of Ostrava, 30 Dubna 22, 701 03, Ostrava 1, Czech Republic

    Florian Luca

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

    Florian Luca

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  1. Harold Erazo
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  2. Carlos A. Gómez
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  3. Florian Luca
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Correspondence to Carlos A. Gómez.

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F. L. was supported in parts by Grant CPRR160325161141 of NRF and the Number Theory Focus Area Grant of CoEMaSS at Wits (South Africa) and CGA 17-02804S (Czech Republic)

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Erazo, H., Gómez, C.A. & Luca, F. Linear combinations of prime powers in X-coordinates of Pell equations. Ramanujan J 53, 123–137 (2020). https://doi.org/10.1007/s11139-019-00213-5

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