A135064 Numbers n such that the quintic polynomial x^5 - 10*n*x^2 - 24*n has Galois group A_5 over rationals.

1, 11, 29, 76, 199, 521, 1364, 3571, 9349, 24476, 64079, 167761, 439204, 1149851, 3010349, 7881196, 20633239, 54018521, 141422324, 370248451, 969323029, 2537720636, 6643838879, 17393796001, 45537549124, 119218851371, 312119004989, 817138163596, 2139295485799, 5600748293801

Offset: 1

Author

Artur Jasinski, Nov 15 2007

Keywords

• nonn

Comments

Sequence appears to agree with the Lucas bisection A002878 for n > 1. - Klaus Brockhaus, Nov 18 2007

A002878(n) is in this sequence for all 1 < n <= 1000, and the sequences agree through a(20) = 370248451. Of course this is not a proof. - Charles R Greathouse IV, Mar 03 2017, updated Mar 20 2017

If this agreement is provable then of course it provides recurrences, generating functions, etc., for this sequence. - N. J. A. Sloane, Nov 24 2007 However, at present this is only a conjecture, and should not be used as the basis for formulas or computer programs. - N. J. A. Sloane, Mar 04 2017

Comparing A135064 with A002878, the number 4 is missing. In this case the Galois group of the quintic polynomial x^5 - 40*x^2 - 96 is dihedral of order 10. - Artur Jasinski, May 27 2010

The relation with A002878 is proved in Wong's article. - Eric M. Schmidt, Nov 25 2017

Links

• Siman Wong, Specialization of Galois groups and integral points on elliptic curves, Proceedings of the American Mathematical Society, 145 (2017), 5179-5190.

Crossrefs

Cf. A134538, A134547, A002878.

Programs

PARI

is(n)=my(p=Pol([1,0,0,-10*n,0,-24*n])); polisirreducible(p) && polgalois(p)[1]==60 \\ Charles R Greathouse IV, Mar 03 2017

Extensions

a(20) corrected by Klaus Brockhaus, Nov 18 2007

Unjustified formulas, programs, and b-file deleted. - N. J. A. Sloane, Mar 04 2017