Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-09T18:53:57.436Z Has data issue: false hasContentIssue false
  • English
  • Français

FACTORIALS AND THE RAMANUJAN FUNCTION

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  21 July 2015

JHON J. BRAVO  and
FLORIAN LUCA
JHON J. BRAVO
Affiliation:
Departamento de Matemáticas, Universidad del Cauca, Calle 5 No 4-70, Popayán, Colombia e-mail: jbravo@unicauca.edu.co
FLORIAN LUCA
Affiliation:
School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa e-mail: Florian.Luca@wits.ac.za
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In 2006, F. Luca and I. E. Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) proved that there are only finitely many pairs (n, m) of positive integers which satisfy the Diophantine equation |τ(n!)|=m!, where τ is the Ramanujan function. In this paper, we follow the same approach of Luca and Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) to determine all solutions of the above equation. The proof of our main theorem uses linear forms in two logarithms and arithmetic properties of the Ramanujan function.

MSC classification

Primary: 11F30: Fourier coefficients of automorphic forms

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 1 , January 2016 , pp. 177 - 185
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

REFERENCES

1.Barkley Rosser, J. and Schoenfeld, L., Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1) (1962), 64–94.Google Scholar
2.Bilu, Y., Hanrot, G. and Voutier, P.M., Existence of primitive divisors of Lucas and Lehmer numbers, with an appendix by M. Mignotte, J. Reine Angew. 539 (2001) (2001), 75–122.Google Scholar
3.Robin, G., Estimation de la fonction de Tchebychef θ sur le k–ième nombre premier et grandes valeurs de la fonction ω(n) nombre de diviseurs premiers de n, (French) (Estimate of the Chebyshev function θ on the kth prime number and large values of the number of prime divisors function ω(n) of n), Acta Arith. 42 (4) (1983), 367–389.CrossRefGoogle Scholar
4.Laurent, M., Mignotte, M. and Nesterenko, Y., Formes linéaires en deux logarithmes et déterminants d'interpolation, (French) (Linear forms in two logarithms and interpolation determinants), J. Number Theory 55 (2) (1995), 285–321.CrossRefGoogle Scholar
5.Luca, F., Equations involving arithmetic functions of factorials, Divulg. Math. 8 (1) (2000), 15–23.Google Scholar
6.Luca, F. and Shparlinski, I. E., Arithmetic properties of the Ramanujan function, Proc. Indian Acad. Sci. (Math. Sci.) 116 (1) (2006), 1–8.CrossRefGoogle Scholar
7.Murty, M. R., The Ramanujan τ function, in Ramanujan revisited: Proceedings of the Centenary Conference (Andrews, G., Editor) (Academic Press, Boston, MA, 1988), 269–288.Google Scholar
8.Stewart, C. L., On divisors of Lucas and Lehmer numbers, Acta Mathematica 211 (2) (2013), 291–314.CrossRefGoogle Scholar