A040038 -id:A040038 - cp's OEIS frontend

A040036 Primes p such that x^3 = 3 has a solution mod p.

2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 61, 67, 71, 73, 83, 89, 101, 103, 107, 113, 131, 137, 149, 151, 167, 173, 179, 191, 193, 197, 227, 233, 239, 251, 257, 263, 269, 271, 281, 293, 307, 311, 317, 347, 353, 359, 367, 383, 389, 401, 419, 431, 439, 443, 449, 461, 467, 479, 491, 499

Offset: 1

N. J. A. Sloane

Complement of A040038 relative to A000040. - Vincenzo Librandi, Sep 13 2012

Being a cube mod p is a necessary condition for 3 to be a 9th power mod p. See Williams link pp. 1, 8 (warning: term 271 is missed). - Michel Marcus, Nov 12 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Kenneth S. Williams, 3 as a Ninth Power (mod p), Math. Scand. Vol 35 (1974), 309-317.

Cf. A000040, A040038. Contains A003627.

[p: p in PrimesUpTo(450) | exists(t){x : x in ResidueClassRing(p) | x^3 eq 3}]; // Vincenzo Librandi, Sep 11 2012

select(p -> isprime(p) and numtheory:-mroot(3,3,p) <> FAIL, [2,seq(i,i=3..1000,2)]); # Robert Israel, Nov 12 2017

ok [p_]:=Reduce[Mod[x^3 - 3, p] == 0, x, Integers] =!= False; Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 11 2012 *)

isok(p) = isprime(p) && ispower(Mod(3,p), 3); \\ Michel Marcus, Nov 12 2017

A107662 -n is the discriminant of cubic polynomials irreducible over Zp for primes p represented by only one binary quadratic form.

Original entry on oeis.org

23, 31, 44, 59, 76, 83, 107, 108, 139, 172, 211, 243, 268, 283, 307, 331, 379, 499, 547, 643, 652, 883, 907

Offset: 1

T. D. Noe, May 19 2005

Let f(x) be any monic integral cubic polynomial with discriminant -n and irreducible over Z. Consider the set S of primes p such that f(x) has no zeros in Zp, i.e., f(x) is irreducible in Zp. For the discriminants -n in this sequence, set S coincides with the primes represented by one binary quadratic form ax^2+bxy+cy^2 with -n=b^2-4ac. For examples, see A106867, A106872, A106282, A106919, A106954, A106967, A040034 and A040038. This sequence consists of (1) terms 4d in A106312 such that the class number of d is 1, (2) terms d in A106312 such that the class number of d is 3 and (3) 108 and 243.

			For each -n, we give (-n,a,b,c) for the quadratic form ax^2+bxy+cy^2: (23,2,1,3), (31,2,1,4), (44,3,2,4), (59,3,1,5), (76,4,2,5), (83,3,1,7), (107,3,1,9), (108,4,2,7), (139,5,1,7), (172,4,2,11), (211,5,3,11), (243,7,3,9), (268,4,2,17), (283,7,5,11), (307,7,1,11), (331,5,3,17), (379,5,1,19), (499,5,1,25), (547,11,5,13), (643,7,1,23), (652,4,2,41), (883,13,1,17) and (907,13,9,19).

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
Blair K. Spearman and Kenneth S. Williams, The cubic congruence x^3+Ax^2+Bx+C = 0 (mod p) and binary quadratic forms, J. London Math. Soc., 46, (1992), 397-410.

Eric Weisstein's World of Mathematics, Class Number

Cf. A106312 (possible negative discriminants of cubic polynomials), A014602 (negative discriminants having class number 1), A006203 (negative discriminants having class number 3).

cp's OEIS Frontend

A040036 Primes p such that x^3 = 3 has a solution mod p.

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