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%I #47 Nov 07 2020 11:42:29
%S 11,23,691
%N Primes p such that tau(p) is congruent to 1 (mod p), where tau is the Ramanujan tau function.
%C 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.
%C a(4) > 500000. - _Dana Jacobsen_, Sep 06 2015
%C a(4) > 10^7. - _Seiichi Manyama_, Nov 25 2017
%C Terms 23 and 691 are exceptional primes for Ramanujan's tau function, see A262339. - _Jud McCranie_, Nov 05 2020
%C A subset of A295645. - _Jud McCranie_, Nov 06 2020
%D M. J. Hopkins, Algebraic topology and modular forms, Proc. Internat. Congress Math., Beijing 2002, Vol. I, pp. 291-317.
%D M. J. Hopkins, Algebraic topology and modular forms, ICM 2002, Vol. I, pp. 283-309.
%H M. J. Hopkins, <a href="http://arxiv.org/abs/math/0212397">Algebraic topology and modular forms</a>, arXiv:math/0212397 [math.AT], 2002.
%H B. Mazur and A. Wiles, <a href="http://www.numdam.org/item?id=CM_1986__59_2_231_0">On p-adic analytic families of Galois representations</a>, Compositio Mathematica, tome 59, n. 2 (1986), p. 231-264.
%t Select[Prime[Range[1, 1000]], 1 == Mod[RamanujanTau[#], #] &] (* _Robert Price_, May 20 2015 *)
%o (Perl) use ntheory ":all"; forprimes { say if (ramanujan_tau($_) % $_) == 1; } 1000; # _Dana Jacobsen_, Sep 06 2015
%o (PARI 2.8) forprime(n=1,1000,if(Mod(tauramanujan(n),n)==1,print1(n,", "))) \\ _Dana Jacobsen_, Sep 06 2015
%Y Cf. A000594, A262339, A295645.
%K nonn,bref,hard,more
%O 1,1
%A _Omar E. Pol_, Aug 14 2011
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