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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.
2

%I #12 May 14 2024 17:17:49

%S 12,64,832,576,4060,86428,8800,76000,17500,61500,22243,303810,60333,

%T 36672,3045440,42588,114244,48552,1251081,486387,579734,209409,

%U 19615484,281216,10826816,406848,378211392,43922220,1051200,1354560,9939228,66545721,773916,9585212

%N 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 Ben Whitmore, <a href="/A371558/b371558.txt">Table of n, a(n) for n = 1..67</a>

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

%e 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 pairs = Join @@ Table[

%t Select[{b, Abs[#1 - b] #2/5} & @@@

%t Sort[SolveValues[x^2 - (6b + 5y^4)x + 25b^2 == 0 && y > 0, {x, y}, Integers]],

%t Max[Last /@ FactorInteger[GCD @@ #]] < 4 &&

%t AllTrue[#, IntegerQ] &&

%t IrreduciblePolynomialQ[x^5 + #1x + #2 & @@ #] &

%t ],

%t {b, -1, -1000, -1}

%t ];

%t pairs[[All, 2]]

%Y For values of b see A371557.

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

%K nonn

%O 1,1

%A _Ben Whitmore_, Apr 22 2024