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 A106867 Primes of the form 2*x^2+x*y+3*y^2. 9
 2, 3, 13, 29, 31, 41, 47, 71, 73, 127, 131, 139, 151, 163, 179, 193, 197, 233, 239, 257, 269, 277, 311, 331, 349, 353, 397, 409, 439, 443, 461, 487, 491, 499, 509, 541, 547, 577, 587, 601, 647, 653, 673, 683, 739, 761, 811, 823, 857, 859, 863, 887, 929, 947 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Discriminant=-23. Primes p such that the polynomial x^3-x-1 is irreducible over Zp. The polynomial discriminant is also -23. - T. D. Noe, May 13 2005 Also, primes p such that tau(p) = A000594(p) == -1 mod 23. [A proof can probably be found in Val der Blij (1952). Thanks to Juan Arias-de-Reyna for this reference. - N. J. A. Sloane, Nov 29 2016 REFERENCES van der Blij, F., Binary quadratic forms of discriminant -23. Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indagationes Math. 14, (1952). 498-503; Math. Rev. MR0052462. Wilton, John Raymond. "Congruence properties of Ramanujan's function τ(n)." Proceedings of the London Mathematical Society 2.1 (1930): 1-10. The primes are listed in Table II. LINKS Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi] D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433. [Annotated scanned copy] N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references) J. R. Wilton, Congruence properties of Ramanujan's function τ(n), annotated copy of page 8 only. MATHEMATICA Union[QuadPrimes2[2, 1, 3, 10000], QuadPrimes2[2, -1, 3, 10000]] (* see A106856 *) PROG (PARI) forprime(p=2, 10^4, if(0==#polrootsmod(x^3-x-1, p), print1(p, ", "))); /* Joerg Arndt, Jul 27 2011 */ (PARI) forprime(p=2, 10^4, if(polisirreducible(Mod(1, p)*(x^3-x-1)), print1(p, ", ") ) ); /* Joerg Arndt, Mar 30 2013 */ CROSSREFS Cf. A086965 (number of distinct zeros of x^3-x-1 mod prime(n)). Cf. also A000594. These are the primes in A028929. Sequence in context: A228991 A141585 A191021 * A141861 A215379 A215375 Adjacent sequences:  A106864 A106865 A106866 * A106868 A106869 A106870 KEYWORD nonn,easy AUTHOR T. D. Noe, May 09 2005 STATUS approved

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Last modified September 20 08:10 EDT 2021. Contains 347577 sequences. (Running on oeis4.)