cp's OEIS Frontend

A316157 Positive integers Q such that there is a cubic x^3 - Qx + R that has three real roots whose continued fraction expansion have common tails.

%I A316157 #26 Jul 31 2018 22:50:07
%S A316157 3,7,9,21,21,39,61,63,93,129,169,171,219,273,331,333,399,471,547,549,
%T A316157 633,723,817,819,921,1029,1141,1143,1263,1389,1519,1521,1659,1803,
%U A316157 1951,1953,2109,2271,2437,2439,2613,2793,2977,2979,3171,3369,3571,3573,3783,3999,4219,4221,4449,4683,4921,4923
%N A316157 Positive integers Q such that there is a cubic x^3 - Qx + R that has three real roots whose continued fraction expansion have common tails.
%C A316157 After 3, the prime terms appear to be the primes in A275878 (namely, 7, 61, 331, 547, 1951, ...)
%H A316157 Joseph-Alfred Serret, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k29114h/f245.image">Section 512</a>, Cours d'algèbre supérieure, Gauthier-Villars.
%e A316157 For the first entry of Q=3, we have the polynomial x^3 - 3x + 1. Its roots, expressed as continued fractions, all have a common tail of 3, 2, 3, 1, 1, 6, 11, ... The next examples are Q=7 with the polynomial x^3 - 7x + 7, then Q=9 with the polynomial x^3 - 9x + 9, and Q=21 with the polynomials x^3 - 21x + 35 and x^3 - 21x + 37. Note that for the Q=7 example, we get the common tail of 2, 3, 1, 6, 10, 5, ... which is contained in A039921.
%t A316157 SetOfQRs = {};
%t A316157 M = 1000;
%t A316157 Do[
%t A316157   If[Divisible[3 (a^2 - a + 1), c^2] &&
%t A316157     Divisible[(2 a - 1) (a^2 - a + 1), c^3] &&
%t A316157     3 (a^2 - a + 1)/c^2 <=  M,
%t A316157    SetOfQRs =
%t A316157     Union[SetOfQRs, { { (3 (a^2 - a + 1))/
%t A316157        c^2, ((2 a - 1) (a^2 - a + 1))/c^3}}   ]],
%t A316157   {c, 1, M/3 + 1, 2}, {a, 1, Sqrt[M c^2/3 + 3/4] + 1/2}];
%t A316157 Print[SetOfQRs // MatrixForm];
%Y A316157 Cf. A316184. Contained in the union of A034017 and three times A034017.
%K A316157 nonn
%O A316157 1,1
%A A316157 _Greg Dresden_, Jun 25 2018
%E A316157 More terms from _Robert G. Wilson v_, Jul 02 2018