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

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

Offset: 1

Author

T. D. Noe, May 09 2005

Keywords

• nonn
• easy

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 van der Blij (1952). Thanks to Juan Arias-de-Reyna for this reference. - N. J. A. Sloane, Nov 29 2016]

References

• 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.
• 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.

Links

• 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.

Crossrefs

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

Cf. also A000594.

These are the primes in A028929.

Programs

Mathematica

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

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 */

Python

from itertools import count, islice
from sympy import prime, GF, Poly
from sympy.abc import x
def A106867_gen(): # generator of terms
    return filter(lambda p:Poly(x**3-x-1,domain=GF(p)).is_irreducible, (prime(i) for i in count(1)))
A106867_list = list(islice(A106867_gen(),20)) # Chai Wah Wu, Nov 11 2022