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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.

%I A371558 #12 May 14 2024 17:17:49
%S A371558 12,64,832,576,4060,86428,8800,76000,17500,61500,22243,303810,60333,
%T A371558 36672,3045440,42588,114244,48552,1251081,486387,579734,209409,
%U A371558 19615484,281216,10826816,406848,378211392,43922220,1051200,1354560,9939228,66545721,773916,9585212
%N 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.
%H A371558 Ben Whitmore, <a href="/A371558/b371558.txt">Table of n, a(n) for n = 1..67</a>
%F A371558 x^5 + A371557(n)*x + a(n) is irreducible and solvable by radicals.
%e A371558 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.
%t A371558 pairs = Join @@ Table[
%t A371558   Select[{b, Abs[#1 - b] #2/5} & @@@
%t A371558     Sort[SolveValues[x^2 - (6b + 5y^4)x + 25b^2 == 0 && y > 0, {x, y}, Integers]],
%t A371558     Max[Last /@ FactorInteger[GCD @@ #]] < 4 &&
%t A371558     AllTrue[#, IntegerQ] &&
%t A371558     IrreduciblePolynomialQ[x^5 + #1x + #2 & @@ #] &
%t A371558   ],
%t A371558   {b, -1, -1000, -1}
%t A371558 ];
%t A371558 pairs[[All, 2]]
%Y A371558 For values of b see A371557.
%Y A371558 For primitive pairs with b > 0 see A371553, A371554.
%K A371558 nonn
%O A371558 1,1
%A A371558 _Ben Whitmore_, Apr 22 2024