A371557 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 b.

-5, -40, -40, -72, -1189, -1189, -1900, -1900, -2625, -2625, -4350, -4350, -7280, -7368, -7368, -7553, -8788, -8840, -8840, -26010, -26010, -29580, -29580, -37180, -37180, -38120, -38120, -43061, -49640, -49640, -63713, -72668, -73185, -73185, -91845, -91845

Offset: 1

Author

Ben Whitmore, Mar 28 2024

Keywords

• sign

Examples

			-40 is in the sequence twice because x^5 - 40*x + 64 and x^5 - 40*x + 832 are both irreducible and solvable by radicals, and (-40, 64) and (-40, 832) are both primitive pairs.
		

Links

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

Crossrefs

For values of c see A371558.

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

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, 1]]

Formula

x^5 + a(n)*x + A371558(n) is irreducible and solvable by radicals.