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

1,1

Comments

After 3, the prime terms appear to be the primes in A275878 (namely, 7, 61, 331, 547, 1951, ...)

Links

Table of n, a(n) for n=1..56.

Joseph-Alfred Serret, Section 512, Cours d'algèbre supérieure, Gauthier-Villars.

Example

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.

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];

Crossrefs

Cf. A316184. Contained in the union of A034017 and three times A034017.

Sequence in context: A304539 A306124 A096102 * A045797 A118555 A056652

Adjacent sequences: A316154 A316155 A316156 * A316158 A316159 A316160

Keyword

nonn

Author

Greg Dresden, Jun 25 2018

Extensions

More terms from Robert G. Wilson v, Jul 02 2018

Status

approved