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.
3, 7, 9, 21, 21, 39, 61, 63, 93, 129, 169, 171, 219, 273, 331, 333, 399, 471, 547, 549, 633, 723, 817, 819, 921, 1029, 1141, 1143, 1263, 1389, 1519, 1521, 1659, 1803, 1951, 1953, 2109, 2271, 2437, 2439, 2613, 2793, 2977, 2979, 3171, 3369, 3571, 3573, 3783, 3999, 4219, 4221, 4449, 4683, 4921, 4923
Offset: 1
Keywords
Examples
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.
Links
- Joseph-Alfred Serret, Section 512, Cours d'algèbre supérieure, Gauthier-Villars.
Programs
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Mathematica
SetOfQRs = {}; M = 1000; Do[ If[Divisible[3 (a^2 - a + 1), c^2] && Divisible[(2 a - 1) (a^2 - a + 1), c^3] && 3 (a^2 - a + 1)/c^2 <= M, SetOfQRs = Union[SetOfQRs, { { (3 (a^2 - a + 1))/ c^2, ((2 a - 1) (a^2 - a + 1))/c^3}} ]], {c, 1, M/3 + 1, 2}, {a, 1, Sqrt[M c^2/3 + 3/4] + 1/2}]; Print[SetOfQRs // MatrixForm];
Extensions
More terms from Robert G. Wilson v, Jul 02 2018
Comments