A033217 Primes of form x^2 + 23*y^2.

23, 59, 101, 167, 173, 211, 223, 271, 307, 317, 347, 449, 463, 593, 599, 607, 691, 719, 809, 821, 829, 853, 877, 883, 991, 997, 1097, 1117, 1151, 1163, 1181, 1231, 1319, 1451, 1453, 1481, 1553, 1613, 1669, 1697, 1787, 1789, 1867, 1871, 1879, 1889, 1913, 2027, 2053, 2143, 2309, 2339, 2347, 2381, 2393, 2423, 2539, 2647, 2677, 2693, 2707, 2741, 2819

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

Discriminant -23.

Also primes of the form x^2 + x*y + 6*y^2. - N. J. A. Sloane, Jun 02 2014

Also primes of the form x^2 - x*y + 6*y^2 with x and y nonnegative. - T. D. Noe, May 07 2005

Primes p such that X^3-X+1 is split modulo p. E.g., X^3-X+1 = (X-33)*(X-40)*(X-94) modulo 167. - Julien Freslon (julien.freslon(AT)wanadoo.fr), Feb 24 2007

It appears that, if x > 0, then tau(p) = A000594(p) == 2 (mod 23). - Comment from Jud McCranie

In fact, this sequence appears to be the same as primes p such that RamanujanTau(p) == {1,2} (mod 23). - Ray Chandler, Dec 01 2016

Excluding the first term, this sequence is the intersection of A191021 and A256567. - Arkadiusz Wesolowski, Oct 03 2021

From Amiram Eldar, Jan 10 2025: (Start)

a(2)..a(10000) are the first terms of the sequence of primes p such that tau(p) == 2 (mod 23), where tau is Ramanujan's tau function (A000594).

Moree and Noubissie (2024) proved that the following 3 conditions for a prime p are equivalent:

1. tau(p) == 2 (mod 23).

2. p divides A000931(p+3) where A000931 is the Padovan sequence.

3. The number of distinct roots modulo p of the polynomial x^3 - x - 1 is 3. (End)

References

• David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
• Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992. See pp. 158-160, "Integer 23 - the Tau function".

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Pieter Moree and Armand Noubissie, Higher reciprocity laws and ternary linear recurrence sequences, International Journal of Number Theory, 2024; arXiv preprint, arXiv:2205.06685 [math.NT], 2022.
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
• 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.
• John Raymond Wilton, Congruence properties of Ramanujan's function τ(n), annotated copy of page 8 only.

Crossrefs

Cf. A000594, A191021, A256567. Primes in A028958.

Programs

Mathematica

QuadPrimes2[1, 0, 23, 10000] (* see A106856 *)
Join[{23}, nn=23; pMax=5000; Union[Reap[Do[p=x^2 + nn y^2; If[p<=pMax&&PrimeQ[p], Sow[p]], {x, Sqrt[pMax]}, {y, Sqrt[pMax/nn]}]][[2, 1]]]] (* Vincenzo Librandi, Sep 05 2016 *)

PARI

isok(p) = isprime(p) && !(kronecker(-23, p)==-1) && !polisirreducible(Mod(1, p)*(x^3-x-1)); \\ Arkadiusz Wesolowski, Oct 03 2021

PARI

isok(p) = p==23 || (isprime(p) && #polrootsmod(x^3-x-1, p)==3); \\ Arkadiusz Wesolowski, Oct 09 2021