A379019 - OEIS

%I #7 Dec 16 2024 08:53:50

%S 21,30,41,48,57,75,84,90,100,102,103,111,129,138,139,152,154,156,165,

%T 183,188,192,201,204,210,219,235,237,246,250,264,269,271,273,291,299,

%U 300,318,327,335,345,348,354,356,372,374,381,384,398,399,404,408,426,433,435,438,446,453,462,480

%N Positive integers k such that the simplest cubic field defined by x^3 - k*x^2 - (k+3)*x - 1 is not monogenic.

%C These are the positive integers k such that the ring of integers O_K of the simplest cubic field K = Q[x]/(x^3 - k*x^2 - (k+3)*x - 1) does not have a power integral basis of the form {1, a, a^2} for any element a in O_K.

%H D. Gil-Muñoz and M. Tinková, <a href="https://arxiv.org/abs/2212.00364">Additive structure of non-monogenic simplest cubic fields</a>, arXiv:2212.00364 [math.NT], 2022.

%H T. Kashio and R. Sekigawa, <a href="https://doi.org/10.2996/kmj44204">The characterization of cyclic cubic fields with power integral bases</a>, Kodai Math. J. 44 (2021), no. 2, 290-306.

%H 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.

%o (Magma)

%o is_A379019 := function(k)

%o     R<x> := PolynomialRing(Integers());

%o     K<a> := NumberField(x^3 - k*x^2 - (k+3)*x - 1);

%o     return #IndexFormEquation(MaximalOrder(K), 1) eq 0;

%o end function;

%o [k : k in [1..1000] | is_A379019(k)];

%Y Cf. A005472.

%K nonn,new

%O 1,1

%A _Robin Visser_, Dec 13 2024