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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/116/01/0001-0008

    • Keywords

       

      Ramanujan τ-function; applications ofS-unit equations

    • Abstract

       

      We study some arithmetic properties of the Ramanujan function τ(n), such as the largest prime divisorP (τ(n)) and the number of distinct prime divisors ω (τ (n)) of τ(n) for various sequences ofn. In particular, we show thatP(τ(n)) ≥ (logn)33/31+o(1) for infinitely many n, and$$P(\tau )(p)\tau (p^2 )\tau (p^3 )) > (1 + o(1))\frac{{\log \log p\log \log \log p}}{{\log \log \log \log p}}$$ for every primep with τ(ρ) ≠ 0.

    • Author Affiliations

       

      Florian Luca1 Igor E Shparlinski2

      1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Morelia, Michoacán - C.P. 58089, México
      2. Department of Computing, Macquarie University, Sydney, NSW - 2109, Australia

    • Dates

       
      Published
      February 2006
      Manuscript received
      2 December 2004

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