The Ramanujan Journal

, Volume 32, Issue 2, pp 269–280 | Cite as

Odd prime values of the Ramanujan tau function

  • Nik Lygeros
  • Olivier Rozier


We study the odd prime values of the Ramanujan tau function, which form a thin set of large primes. To this end, we define LR(p,n):=τ(p n−1) and we show that the odd prime values are of the form LR(p,q) where p,q are odd primes. Then we exhibit arithmetical properties and congruences of the LR numbers using more general results on Lucas sequences. Finally, we propose estimations and discuss numerical results on pairs (p,q) for which LR(p,q) is prime.


Ramanujan function Primality Lucas sequences 

Mathematics Subject Classification (2010)

11A41 11F30 11Y11 



We are grateful to François Morain for his outstanding contribution to the numerical results. We also would like to thank Marc Hufschmitt and Paul Zimmermann for their helpful discussions, and the anonymous referee for his valuable suggestions.


  1. 1.
    Bilu, Y., Hanrot, G., Voutier, P.M.: Existence of primitive divisors of Lucas and Lehmer numbers. J. Reine Angew. Math. 539, 75–122 (2001) MathSciNetMATHGoogle Scholar
  2. 2.
    Lehmer, D.H.: The primality of Ramanujan’s Tau-function. Am. Math. Mon. 72, 15–18 (1965) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Lucas, E.: Théorie des fonctions numériques simplement périodiques. Am. J. Math. 1, 184–240 and 289–321 (1878) CrossRefMATHGoogle Scholar
  4. 4.
    Lygeros, N., Rozier, O.: A new solution for the equation τ(p)≡0(modp). J. Integer Seq. 13(10.7.4), 1–11 (2010) MathSciNetGoogle Scholar
  5. 5.
    Morain, F.: Implementing the asymptotically fast version of the elliptic curve primality proving algorithm. Math. Comput. 76, 493–505 (2007) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Murty, M.R., Murty, V.K., Shorey, T.N.: Odd values of the Ramanujan τ-function. Bull. Soc. Math. Fr. 115, 391–395 (1987) MathSciNetMATHGoogle Scholar
  7. 7.
    Ramanujan, S.: On certain arithmetical functions. Trans. Camb. Philos. Soc. 22, 159–184 (1916) Google Scholar
  8. 8.
    Ramanujan, S.: A proof of Bertrand’s postulate. J. Indian Math. Soc. 11, 181–182 (1919) Google Scholar
  9. 9.
    Ribenboim, P.: The New Book of Prime Number Records. Springer, Berlin (1996) CrossRefMATHGoogle Scholar
  10. 10.
    Serre, J.-P.: Divisibilité de certaines fonctions arithmétiques. Enseign. Math. 22, 227–260 (1976) MATHGoogle Scholar
  11. 11.
    Swinnerton-Dyer, H.P.F.: On -adic representations and congruences for coefficients of modular forms. Lect. Notes Math. 350, 1–55 (1973) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Wagstaff, S.S.: Divisors of Mersenne numbers. Math. Comput. 40, 385–397 (1983) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  1. 1.LGPC (UMR 5285)Université de LyonVilleurbanneFrance
  2. 2.Service de calcul parallèle S-CAPADIPGP (UMR 7154)ParisFrance

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