A193855 Primes p such that tau(p) is congruent to 1 (mod p), where tau is the Ramanujan tau function.

11, 23, 691

Offset: 1

Author

Omar E. Pol, Aug 14 2011

Keywords

• nonn
• bref
• hard
• more

Comments

M. J. Hopkins wrote "It is not known whether or not tau(p) == 1 mod p holds for infinitely many primes". For more information about this open problem see the Sloane comment in A000594.

a(4) > 500000. - Dana Jacobsen, Sep 06 2015

a(4) > 10^7. - Seiichi Manyama, Nov 25 2017

Terms 23 and 691 are exceptional primes for Ramanujan's tau function, see A262339. - Jud McCranie, Nov 05 2020

A subset of A295645. - Jud McCranie, Nov 06 2020

References

• M. J. Hopkins, Algebraic topology and modular forms, Proc. Internat. Congress Math., Beijing 2002, Vol. I, pp. 291-317.
• M. J. Hopkins, Algebraic topology and modular forms, ICM 2002, Vol. I, pp. 283-309.

Links

• M. J. Hopkins, Algebraic topology and modular forms, arXiv:math/0212397 [math.AT], 2002.
• B. Mazur and A. Wiles, On p-adic analytic families of Galois representations, Compositio Mathematica, tome 59, n. 2 (1986), p. 231-264.

Crossrefs

Cf. A000594, A262339, A295645.

Programs

Mathematica

Select[Prime[Range[1, 1000]], 1 == Mod[RamanujanTau[#], #] &] (* Robert Price, May 20 2015 *)

PARI

forprime(n=1,1000,if(Mod(ramanujantau(n),n)==1,print1(n,", "))) \\ Dana Jacobsen, Sep 06 2015

Perl

use ntheory ":all"; forprimes { say if (ramanujan_tau($) % $) == 1; } 1000; # Dana Jacobsen, Sep 06 2015