A275879
Nonstandard Jacobi primes.
Original entry on oeis.org
13, 97, 193, 409, 769, 2593, 4729, 6481, 12289, 15361, 55681, 331777, 417793, 737281, 786433, 2752513, 6684673, 8650753, 36175873, 69206017, 75079681, 155344897, 270532609, 435486721, 824717353, 860301577, 1380974593, 1845657601, 3221225473, 3255828481, 3281584129, 8531146753
Offset: 1
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.
Original entry on oeis.org
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
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.
- Joseph-Alfred Serret, Section 512, Cours d'algèbre supérieure, Gauthier-Villars.
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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];
Showing 1-2 of 2 results.
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