A227622 Primes p of the form m^2 + 27.

31, 43, 127, 223, 283, 811, 1051, 1471, 1627, 2143, 2731, 3163, 3391, 4651, 5503, 6427, 8863, 9631, 16411, 16927, 18523, 23131, 23743, 27583, 28927, 29611, 33151, 37663, 42463, 43291, 44971, 45823, 56671, 65563, 70783, 78427, 80683, 84127, 87643, 106303, 110251, 122527, 123931, 131071

Offset: 1

Author

William P. Orrick, Jul 17 2013

Keywords

• nonn
• easy

Comments

Orders for which residues mod p of the form x^i, i congruent to 0, 1, or 3 (mod 6), form a difference set with parameters (v,k,lambda)=(p,(p-1)/2,(p-3)/4), where x is a primitive root such that 3=x^j, with j congruent to 1 (mod 6). This construction is due to Marshall Hall. Such a difference set has the same parameters as the difference set formed by quadratic residues, that is, the Paley difference set, but is not equivalent to it. Both difference sets give rise to Hadamard matrices of size p+1.

From Peter Bala, Nov 19 2021: (Start)

2 is a cube mod p (a particular case of a more general result of Gauss). See A014752.

Primes of the form a^2 + 6*a + 36, where a is an integer.

Let p == 1 (mod 6) be a prime. There are integers c and d, unique up to sign, such that 4*p = c^2 + 27*d^2 [see, for example, Ireland and Rosen, Proposition 8.3.2]. This sequence lists those primes with d = 2. Cf. A005471 (case d = 1) and A349461 (case d = 3).

Primes p of the form m^2 + 27 are related to cyclic cubic fields in several ways:

(1) The cubic polynomial X^3 - p*X + 2*p, with discriminant 4*m^2*p^2, a square, is irreducible over Q by Eisenstein's criteria. It follows that the Galois group of the polynomial over Q is the cyclic group C_3 (apply Conrad, Corollary 2.5).

Note that the roots of the cubic X^3 - p*X + 2*p, are the differences n_0 - n_1, n_1 - n_2 and n_2 - n_0 of the cubic Gaussian periods n_i for the modulus p.

(2) The cubic 2*X^3 + p*(X + 2)^2, with discriminant 64*m^2*p^2, a square, is irreducible over Q by Eisenstein's criteria, and so the Galois group of the polynomial over Q is the cyclic group C_3.

(3) The cubic X^3 - (m-3)*X^2 - 2*(m+3)*X - 8, has discriminant (2*p)^2, a square. (This is the polynomial g_3(m-3, 0, -2; X) in the notation of Hashimoto and Hoshi.) The cubic is irreducible over Q for nonzero m by the Rational Root Theorem and hence the Galois group of the polynomial over Q is the cyclic group C_3. (End)

Examples

			For p=31, using x=3 as primitive root, the set of residues {1,2,3,4,6,8,12,15,16,17,23,24,27,29,30} is a difference set.
2 a cube mod p: 4^3 == 2 (mod 31); 20^3 == 2 (mod 43); 8^3 == 2 (mod 127); 68^3 == 2 (mod 223). - _Peter Bala_, Nov 19 2021
		

References

• K. Ireland and M. Rosen, A classical introduction to modern number theory, vol. 84, Graduate Texts in Mathematics. Springer-Verlag. [Prop. 8.3.2, p. 96]
• Thomas Storer, Cyclotomy and difference sets. Markham, Chicago, 1967, pages 73-76.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Peter Bala, Notes on the period polynomial for the cubic Gaussian periods
• Keith Conrad, Galois groups of cubics and quartics (not in characteristic 2)
• Marshall Hall Jr., A survey of difference sets, Proc. Amer. Math. Soc. 7 (1956) 975-986.
• Ki-Ichiro Hashimoto and Akinari Hoshi, Families of cyclic polynomials obtained from geometric generalization of Gaussian period relations, Math. Comp., Vol. 74, No. 251, 2005, pp. 1519-1530
• D. H. Lehmer and Emma Lehmer, The Lehmer Project, Math. of Comp., Vol. 61, No. 203, 1993, pp. 313-317.

Crossrefs

Cf. A005471, A014752, A349461.

Programs

Mathematica

Select[Table[m^2+27,{m,0,100}],PrimeQ]

PARI

for(n=0,10^3,my(p=n^2+27);if(isprime(p),print1(p,", "))); \\ Joerg Arndt, Jul 18 2013