OFFSET
1,1
COMMENTS
3 is a cube mod p for all primes in this list; this is a particular case of a result of Gauss. See Ireland and Rosen, Chapter 9, Exercise 23, p. 135. Some examples are given below.
Primes p such that 4*p - 243 is a square. Let p == 1 (mod 6) be a prime. There are integers c and d such that 4*p = c^2 + 27*d^2 (see, for example, Ireland and Rosen, Proposition 8.3.2). This sequence lists the primes with d = 3. Cf. A005471 (case d = 1) and A227622 (case d = 2).
Primes p of the form m^2 + 9*m + 81 are related to cyclic cubic fields in several ways:
(1) The cubic x^3 - p*x + 3*p, with discriminant ((2*m + 9)*p)^2, 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 x^3 - p*x + 3*p are the differences n_0 - n_1, n_1 - n_2 and n_2 - n_0, where n_0, n_1 and n_2 are the three cubic Gaussian periods for the modulus p.
(2) The cubic x^3 - m*x^2 - 3*(m + 9)*x - 27 has discriminant (3*p)^2, a square. This is the polynomial g_3(a, 0, -3; X) in the notation of Hashimoto and Hoshi. The cubic is irreducible over Q by the Rational Root Theorem and hence the Galois group of the polynomial over Q is the cyclic group C_3.
(3) The cubic 3*x^3 + p*(x + 3)^2, with discriminant 81*p^2*(4*p - 243), 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.
REFERENCES
K. Ireland and M. Rosen, A classical introduction to modern number theory, Vol. 84, Graduate Texts in Mathematics, Springer-Verlag.
LINKS
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.
D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152
EXAMPLE
61 = (-4)^2 + 9*(-4) + 81; 67 = (-2)^2 + 9*(-2) + 81; 73 = (-1)^2 + 9*(-1) + 81; 103 = (2)^2 + 9*(2) + 81.
3 is a cube mod p:
4^3 == 3 (mod 61); 18^3 == 3 (mod 67); 25^3 == 3 (mod 73); 67^3 == 3 (mod 103).
MATHEMATICA
Select[Table[m^2+9*m+81, {m, -4, 120}], PrimeQ]
PROG
(PARI) for (m = -4, 120, my(p = m^2 + 9*m + 81); if (isprime(p), print1(p, ", ")));
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Nov 18 2021
STATUS
approved