Functiones et Approximatio Commentarii Mathematici

Lucas non-Wieferich primes in arithmetic progressions

Sudhansu Sekhar Rout

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this note, we define the Lucas Wieferich primes which are an analogue of the famous Wieferich primes. Conditionally there are infinitely many non-Wieferich primes. We prove under the assumption of the $abc$ conjecture for the number field $\mathbb{Q}(\sqrt{\Delta})$ that for fixed positive integer~$M$ there are at least $O((\log x/\log \log x)(\log \log \log x)^{M})$ many Lucas non-Wieferich primes $p \equiv 1(mod k)$ for any fixed integer $k\geq 2$.

Article information

Source
Funct. Approx. Comment. Math., Volume 60, Number 2 (2019), 167-175.

Dates
First available in Project Euclid: 29 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.facm/1543460427

Digital Object Identifier
doi:10.7169/facm/1709

Mathematical Reviews number (MathSciNet)
MR3964258

Zentralblatt MATH identifier
07068529

Subjects
Primary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations 11B25: Arithmetic progressions [See also 11N13] 11A41: Primes

Keywords
Lucas-Wieferich primes arithmetic progressions $abc$ conjecture

Citation

Rout, Sudhansu Sekhar. Lucas non-Wieferich primes in arithmetic progressions. Funct. Approx. Comment. Math. 60 (2019), no. 2, 167--175. doi:10.7169/facm/1709. https://projecteuclid.org/euclid.facm/1543460427


Export citation

References

  • [1] Y. Bilu, G. Hanrot and P.M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, with an appendix by M. Mignotte, J. Reine Angew. Math. 539 (2001), 75–122.
  • [2] R.D. Carmichael, On the numerical factors of the arithmetic forms $\alpha^n \pm \beta^n$, Ann. of Math. (2) 15(1–4) (1913/14), 30–70.
  • [2] Y. Chen and Y. Ding, Non-Wieferich primes in arithmetic progressions, Proc. Amer. Math. Soc. 145 (2017), 1833–1836.
  • [3] L.E. Dickson, The History of the Theory of Numbers, vol. 1, reprinted: Chelsea Publishing Company, New York, 1966.
  • [4] J.M. Dekonick and N. Doyon, On the set of Wieferich primes and its complement, Anna. Univ. Sci. Budapest, Sect. Comput. 27 (2007), 3–13.
  • [5] G. Everest, A. van der Poorten, I.E. Shparlinski and T. Ward, Recurrence sequences, Mathematical Surveys and Monographs, Volume 104, American Mathematical Society, Providence, RI, 2003.
  • [6] A. Granville, Powerful numbers and Fermat's last theorem, C.R. Math. Rep. Acad. Sc. Canada 8 (1986), 215–218.
  • [7] A. Granville and M. Monagan, The first case of Fermat's last theorem is true for all prime exponents up to $714, 591, 416, 091, 389$, Trans. Amer. Math. Soc. 306 (1988), 329–359.
  • [8] H. Graves and M.R. Murty, The $abc$ conjecture and non-Wieferic primes in arithmetic progressions, J. Number Theory 133 (2013), 1809–1813.
  • [9] K. Gy\Hory, On the $abc$ conjecture in algebraic number fields, Acta. Arith. 133 (2008), 281–295.
  • [10] G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 5th edition, The Clarendon Press, Oxford University Press, New York, 1979.
  • [11] R.J. McIntosh and E.L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp. 76 (2007), 2087–2094.
  • [12] M.R. Murty, Problems in Analytic Number Theory, second edition, Grad. Texts. in Math., Springer, 2008.
  • [13] P. Ribenboim, On square factors of terms of binary recurring sequences and the $abc$ conjecture, Publ. Math. Debrecen 59 (2001), 459–469.
  • [14] J.B. Rosser, The $n$-th prime is greater than $n\log n$, Proc. London Math. Soc. 45 (1938), 21–44.
  • [15] S.S. Rout, Balancing non-Wieferich primes in arithmetic progression and $abc$ conjecture, Proc. Japan Acad. Ser. A Math. Sci. 92(9) (2016), 112–116.
  • [16] J. Silverman, Wieferich's criterion and the abc conjecture, J. Number Theory 30 (1988), 226–237.
  • [17] L. Somer, Divisibility of terms in Lucas sequences by their subscripts, in Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), pages 515–525, Kluwer Acad. Publ., Dordrecht, 1993.
  • [18] C.L. Stewart, On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers, Proc. London Math. Soc. 35 (1977), 425–447.
  • [19] Z.-H. Sun and Z.-W. Sun, Fibonacci numbers and Fermat's last theorem, Acta. Arith. 60 (1992), 371–388.
  • [20] P. Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, Springer-Verlag, 1980.
  • [21] A. Wieferich, Zum letzten Fermatschen Theorem (German), J. Reine Angew. Math 136 (1909), 293–302.