cp's OEIS Frontend

A349461 Primes of the form m^2 + 9*m + 81.

%I A349461 #15 Nov 24 2021 08:17:32
%S A349461 61,67,73,103,151,193,271,367,523,613,661,991,1117,1321,1543,1621,
%T A349461 1783,1867,2131,2713,3253,3967,4093,4483,6067,6703,7717,8803,9181,
%U A349461 10567,11617,11833,13171,13633,14341,15313
%N A349461 Primes of the form m^2 + 9*m + 81.
%C A349461 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.
%C A349461 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).
%C A349461 Primes p of the form m^2 + 9*m + 81 are related to cyclic cubic fields in several ways:
%C A349461 (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).
%C A349461 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.
%C A349461 (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.
%C A349461 (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.
%D A349461 K. Ireland and M. Rosen, A classical introduction to modern number theory, Vol. 84, Graduate Texts in Mathematics, Springer-Verlag.
%H A349461 Peter Bala, <a href="/A349461/a349461.pdf">Notes on the period polynomial for the cubic Gaussian periods</a>
%H A349461 Keith Conrad, <a href="http://www.ms.uky.edu/~sohum/ma561/notes/workspace/books/cubicquartic.pdf">Galois groups of cubics and quartics (not in characteristic 2)</a>
%H A349461 Ki-Ichiro Hashimoto and Akinari Hoshi, <a href="https://doi.org/10.1090/S0025-5718-05-01750-3">Families of cyclic polynomials obtained from geometric generalization of Gaussian period relations</a>, Math. Comp., Vol. 74, No. 251, 2005, pp. 1519-1530
%H A349461 D. H. Lehmer and Emma Lehmer, <a href="https://doi.org/10.2307/2152956">The Lehmer Project</a>, Math. of Comp., Vol. 61, No. 203, 1993, pp. 313-317.
%H A349461 D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1974-0352049-8">The simplest cubic fields</a>, Math. Comp., 28 (1974), 1137-1152
%e A349461 61 = (-4)^2 + 9*(-4) + 81; 67 = (-2)^2 + 9*(-2) + 81; 73 = (-1)^2 + 9*(-1) + 81; 103 = (2)^2 + 9*(2) + 81.
%e A349461 3 is a cube mod p:
%e A349461 4^3 == 3 (mod 61); 18^3 == 3 (mod 67); 25^3 == 3 (mod 73); 67^3 == 3 (mod 103).
%t A349461 Select[Table[m^2+9*m+81, {m, -4, 120}], PrimeQ]
%o A349461 (PARI) for (m = -4, 120, my(p = m^2 + 9*m + 81); if (isprime(p), print1(p,", ")));
%Y A349461 Cf. A014753, A005471, A227622.
%K A349461 nonn,easy
%O A349461 1,1
%A A349461 _Peter Bala_, Nov 18 2021