cp's OEIS Frontend

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

%I A106867 #49 Nov 12 2022 03:08:45
%S A106867 2,3,13,29,31,41,47,71,73,127,131,139,151,163,179,193,197,233,239,257,
%T A106867 269,277,311,331,349,353,397,409,439,443,461,487,491,499,509,541,547,
%U A106867 577,587,601,647,653,673,683,739,761,811,823,857,859,863,887,929,947
%N A106867 Primes of the form 2*x^2 + x*y + 3*y^2.
%C A106867 Discriminant = -23.
%C A106867 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 A106867 Also, primes p such that tau(p) = A000594(p) == -1 (mod 23). [A proof can probably be found in van der Blij (1952). Thanks to _Juan Arias-de-Reyna_ for this reference. - _N. J. A. Sloane_, Nov 29 2016]
%D A106867 F. van der Blij, Binary quadratic forms of discriminant -23. Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indagationes Math. 14, (1952). 498-503; Math. Rev. MR0052462.
%D A106867 John Raymond Wilton, "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 A106867 Ray Chandler, <a href="/A106867/b106867.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Vincenzo Librandi)
%H A106867 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 A106867 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 A106867 J. R. Wilton, <a href="/A278578/a278578.pdf">Congruence properties of Ramanujan's function τ(n)</a>, annotated copy of page 8 only.
%t A106867 Union[QuadPrimes2[2, 1, 3, 10000], QuadPrimes2[2, -1, 3, 10000]] (* see A106856 *)
%o A106867 (PARI) forprime(p=2,10^4,if(0==#polrootsmod(x^3-x-1,p),print1(p,", "))); /* _Joerg Arndt_, Jul 27 2011 */
%o A106867 (PARI) forprime(p=2,10^4,if(polisirreducible(Mod(1, p)*(x^3-x-1)), print1(p, ", ") ) ); /* _Joerg Arndt_, Mar 30 2013 */
%o A106867 (Python)
%o A106867 from itertools import count, islice
%o A106867 from sympy import prime, GF, Poly
%o A106867 from sympy.abc import x
%o A106867 def A106867_gen(): # generator of terms
%o A106867     return filter(lambda p:Poly(x**3-x-1,domain=GF(p)).is_irreducible, (prime(i) for i in count(1)))
%o A106867 A106867_list = list(islice(A106867_gen(),20)) # _Chai Wah Wu_, Nov 11 2022
%Y A106867 Cf. A086965 (number of distinct zeros of x^3-x-1 mod prime(n)).
%Y A106867 Cf. also A000594.
%Y A106867 These are the primes in A028929.
%K A106867 nonn,easy
%O A106867 1,1
%A A106867 _T. D. Noe_, May 09 2005