%I #11 Jul 31 2018 22:49:31

%S 1,7,9,35,37,91,183,189,341,559,845,855

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

%H Joseph-Alfred Serret, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k29114h/f245.image">Section 512</a>, Cours d'algèbre supérieure, Gauthier-Villars.

%e For the first entry of R=1, 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 R=7 with the polynomial x^3 - 7x + 7, then R=9 with the polynomial x^3 - 9x + 9, and Q=35 with the polynomial x^3 - 21x + 35. Note that for the R=7 example, we get the common tail of 2, 3, 1, 6, 10, 5, ... which is contained in A039921.

%t SetOfQRs = {};

%t M = 1000;

%t Do[

%t If[Divisible[3 (a^2 - a + 1), c^2] &&

%t Divisible[(2 a - 1) (a^2 - a + 1), c^3] &&

%t 3 (a^2 - a + 1)/c^2 <= M,

%t SetOfQRs =

%t Union[SetOfQRs, { { (3 (a^2 - a + 1))/

%t c^2, ((2 a - 1) (a^2 - a + 1))/c^3}} ]],

%t {c, 1, M/3 + 1, 2}, {a, 1, Sqrt[M c^2/3 + 3/4] + 1/2}];

%t Print[SetOfQRs // MatrixForm];

%Y Cf. A039921, A316157.

%K nonn,more

%O 1,2

%A _Greg Dresden_, Jun 25 2018