cp's OEIS Frontend

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

%I A033217 #64 Jan 11 2025 02:41:17
%S A033217 23,59,101,167,173,211,223,271,307,317,347,449,463,593,599,607,691,
%T A033217 719,809,821,829,853,877,883,991,997,1097,1117,1151,1163,1181,1231,
%U A033217 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
%N A033217 Primes of form x^2 + 23*y^2.
%C A033217 Discriminant -23.
%C A033217 Also primes of the form x^2 + x*y + 6*y^2. - _N. J. A. Sloane_, Jun 02 2014
%C A033217 Also primes of the form x^2 - x*y + 6*y^2 with x and y nonnegative. - _T. D. Noe_, May 07 2005
%C A033217 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
%C A033217 It appears that, if x > 0, then tau(p) = A000594(p) == 2 (mod 23). - Comment from _Jud McCranie_
%C A033217 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
%C A033217 Excluding the first term, this sequence is the intersection of A191021 and A256567. - _Arkadiusz Wesolowski_, Oct 03 2021
%C A033217 From _Amiram Eldar_, Jan 10 2025: (Start)
%C A033217 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).
%C A033217 Moree and Noubissie (2024) proved that the following 3 conditions for a prime p are equivalent:
%C A033217   1. tau(p) == 2 (mod 23).
%C A033217   2. p divides A000931(p+3) where A000931 is the Padovan sequence.
%C A033217   3. The number of distinct roots modulo p of the polynomial x^3 - x - 1 is 3. (End)
%D A033217 David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
%D A033217 Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992. See pp. 158-160, "Integer 23 - the Tau function".
%H A033217 Ray Chandler, <a href="/A033217/b033217.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A033217 Pieter Moree and Armand Noubissie, <a href="https://doi.org/10.1142/S1793042125500502">Higher reciprocity laws and ternary linear recurrence sequences</a>, International Journal of Number Theory, 2024; <a href="https://arxiv.org/abs/2205.06685">arXiv preprint</a>, arXiv:2205.06685 [math.NT], 2022.
%H A033217 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 A033217 John Raymond Wilton, <a href="https://doi.org/10.1112/plms/s2-31.1.1">Congruence properties of Ramanujan's function τ(n)</a>, Proceedings of the London Mathematical Society 2.1 (1930): 1-10. The primes are listed in Table II.
%H A033217 John Raymond Wilton, <a href="/A278578/a278578.pdf">Congruence properties of Ramanujan's function τ(n)</a>, annotated copy of page 8 only.
%t A033217 QuadPrimes2[1, 0, 23, 10000] (* see A106856 *)
%t A033217 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 *)
%o A033217 (PARI) isok(p) = isprime(p) && !(kronecker(-23, p)==-1) && !polisirreducible(Mod(1, p)*(x^3-x-1)); \\ _Arkadiusz Wesolowski_, Oct 03 2021
%o A033217 (PARI) isok(p) = p==23 || (isprime(p) && #polrootsmod(x^3-x-1, p)==3); \\ _Arkadiusz Wesolowski_, Oct 09 2021
%Y A033217 Cf. A000594, A191021, A256567. Primes in A028958.
%K A033217 nonn,nice
%O A033217 1,1
%A A033217 _N. J. A. Sloane_