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A371557
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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 b.
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2
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-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
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OFFSET
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1,1
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LINKS
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FORMULA
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x^5 + a(n)*x + A371558(n) is irreducible and solvable by radicals.
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EXAMPLE
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-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.
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MATHEMATICA
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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]]
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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