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

1,1

Links

Ben Whitmore, Table of n, a(n) for n = 1..67

Formula

x^5 + A371557(n)*x + a(n) is irreducible and solvable by radicals.

Example

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.

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]]

Crossrefs

For values of b see A371557.

For primitive pairs with b > 0 see A371553, A371554.

Sequence in context: A307061 A074359 A104062 * A232383 A003868 A304833

Adjacent sequences: A371555 A371556 A371557 * A371559 A371560 A371561

Keyword

nonn

Author

Ben Whitmore, Apr 22 2024

Status

approved