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 A106867 Primes of the form 2*x^2+x*y+3*y^2. 9

%I

%S 2,3,13,29,31,41,47,71,73,127,131,139,151,163,179,193,197,233,239,257,

%T 269,277,311,331,349,353,397,409,439,443,461,487,491,499,509,541,547,

%U 577,587,601,647,653,673,683,739,761,811,823,857,859,863,887,929,947

%N Primes of the form 2*x^2+x*y+3*y^2.

%C Discriminant=-23.

%C 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

%C 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

%D 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.

%D 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.

%H Vincenzo Librandi and Ray Chandler, <a href="/A106867/b106867.txt">Table of n, a(n) for n = 1..10000</a> [First 1000 terms from Vincenzo Librandi]

%H D. H. Lehmer, <a href="/A000594/a000594.pdf">The Vanishing of Ramanujan's Function tau(n)</a>, Duke Mathematical Journal, 14 (1947), pp. 429-433. [Annotated scanned copy]

%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)

%H J. R. Wilton, <a href="/A278578/a278578.pdf">Congruence properties of Ramanujan's function τ(n)</a>, annotated copy of page 8 only.

%t Union[QuadPrimes2[2, 1, 3, 10000], QuadPrimes2[2, -1, 3, 10000]] (* see A106856 *)

%o (PARI) forprime(p=2,10^4,if(0==#polrootsmod(x^3-x-1,p),print1(p,", "))); /* _Joerg Arndt_, Jul 27 2011 */

%o (PARI) forprime(p=2,10^4,if(polisirreducible(Mod(1, p)*(x^3-x-1)), print1(p, ", ") ) ); /* _Joerg Arndt_, Mar 30 2013 */

%Y Cf. A086965 (number of distinct zeros of x^3-x-1 mod prime(n)).

%Y Cf. also A000594.

%Y These are the primes in A028929.

%K nonn,easy

%O 1,1

%A _T. D. Noe_, May 09 2005

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Last modified November 21 04:41 EST 2019. Contains 329350 sequences. (Running on oeis4.)