A316184 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.
1, 7, 9, 35, 37, 91, 183, 189, 341, 559, 845, 855
Offset: 1
Examples
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.
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];