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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.

%I A371557 #16 May 14 2024 16:59:39
%S A371557 -5,-40,-40,-72,-1189,-1189,-1900,-1900,-2625,-2625,-4350,-4350,-7280,
%T A371557 -7368,-7368,-7553,-8788,-8840,-8840,-26010,-26010,-29580,-29580,
%U A371557 -37180,-37180,-38120,-38120,-43061,-49640,-49640,-63713,-72668,-73185,-73185,-91845,-91845
%N 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.
%H A371557 Ben Whitmore, <a href="/A371557/b371557.txt">Table of n, a(n) for n = 1..67</a>
%F A371557 x^5 + a(n)*x + A371558(n) is irreducible and solvable by radicals.
%e A371557 -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.
%t A371557 pairs = Join @@ Table[
%t A371557   Select[{b, Abs[#1 - b] #2/5} & @@@
%t A371557     Sort[SolveValues[x^2 - (6b + 5y^4)x + 25b^2 == 0 && y > 0, {x, y}, Integers]],
%t A371557     Max[Last /@ FactorInteger[GCD @@ #]] < 4 &&
%t A371557     AllTrue[#, IntegerQ] &&
%t A371557     IrreduciblePolynomialQ[x^5 + #1x + #2 & @@ #] &
%t A371557   ],
%t A371557   {b, -1, -1000, -1}
%t A371557 ];
%t A371557 pairs[[All, 1]]
%Y A371557 For values of c see A371558.
%Y A371557 For primitive pairs with b > 0 see A371553, A371554.
%K A371557 sign
%O A371557 1,1
%A A371557 _Ben Whitmore_, Mar 28 2024