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

1,1

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A274381 A044125 A044506 * A142107 A107208 A289735

Adjacent sequences: A033214 A033215 A033216 * A033218 A033219 A033220

Keyword

nonn,nice

Author

N. J. A. Sloane

Status

approved