A371558 Consider primitive pairs of integers (b, c) with b < 0 such that x^5 + b*x + c = 0 is irreducible and solvable by radicals: sequence gives values of c.
12, 64, 832, 576, 4060, 86428, 8800, 76000, 17500, 61500, 22243, 303810, 60333, 36672, 3045440, 42588, 114244, 48552, 1251081, 486387, 579734, 209409, 19615484, 281216, 10826816, 406848, 378211392, 43922220, 1051200, 1354560, 9939228, 66545721, 773916, 9585212
Offset: 1
Keywords
Examples
a(1) = 12 because A371557(1) = -5, and x^5 - 5*x + 12 is irreducible and solvable by radicals, and (-5, 12) is a primitive pair.
Links
- Ben Whitmore, Table of n, a(n) for n = 1..67
Programs
-
Mathematica
pairs = Join @@ Table[ Select[{b, Abs[#1 - b] #2/5} & @@@ Sort[SolveValues[x^2 - (6b + 5y^4)x + 25b^2 == 0 && y > 0, {x, y}, Integers]], Max[Last /@ FactorInteger[GCD @@ #]] < 4 && AllTrue[#, IntegerQ] && IrreduciblePolynomialQ[x^5 + #1x + #2 & @@ #] & ], {b, -1, -1000, -1} ]; pairs[[All, 2]]
Formula
x^5 + A371557(n)*x + a(n) is irreducible and solvable by radicals.