login
A193855
Primes p such that tau(p) is congruent to 1 (mod p), where tau is the Ramanujan tau function.
3
OFFSET
1,1
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.
MATHEMATICA
Select[Prime[Range[1, 1000]], 1 == Mod[RamanujanTau[#], #] &] (* Robert Price, May 20 2015 *)
PROG
(Perl) use ntheory ":all"; forprimes { say if (ramanujan_tau($_) % $_) == 1; } 1000; # Dana Jacobsen, Sep 06 2015
(PARI) forprime(n=1, 1000, if(Mod(ramanujantau(n), n)==1, print1(n, ", "))) \\ Dana Jacobsen, Sep 06 2015
CROSSREFS
KEYWORD
nonn,bref,hard,more
AUTHOR
Omar E. Pol, Aug 14 2011
STATUS
approved