A371554 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.

44, 12, 44, 32, 64, 1344, 576, 1344, 832, 275, 4170, 2375, 3750, 4060, 128700, 13243, 1510620, 24000, 3348800, 8788, 467961, 51072, 133440, 474214, 61500, 128700, 85683, 514098, 509197, 199927, 24000, 3720000, 21376538, 210990, 486343, 114244, 12681084

Offset: 1

Author

Ben Whitmore, Mar 27 2024

Keywords

• nonn

Comments

See A371553.

Examples

			a(1) = 44 because A371553(1) = 11, and x^5 + 11*x + 44 is irreducible and solvable by radicals, and (11, 44) is a primitive pair.
		

Links

• Ben Whitmore, Table of n, a(n) for n = 1..95

Crossrefs

For values of b see A371553.

Programs

Mathematica

pairs = Join @@ Table[
  Select[{m, 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}
];
pairs[[All, 2]]

Formula

x^5 + A371553(n)*x + a(n) is irreducible and solvable by radicals.