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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
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Abstract

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

References

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