A371554
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.
Original entry on oeis.org
44, 12, 44, 32, 64, 1344, 576, 1344, 832, 275, 4170, 2375, 3750, 4060, 128700, 13243, 1510620, 24000, 3348800, 8788, 467961, 51072, 133440, 474214, 61500, 128700, 85683, 514098, 509197, 199927, 24000, 3720000, 21376538, 210990, 486343, 114244, 12681084
Offset: 1
a(1) = 44 because A371553(1) = 11, and x^5 + 11*x + 44 is irreducible and solvable by radicals, and (11, 44) is a primitive pair.
-
pairs = Join @@ Table[
Select[{m, 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}
];
pairs[[All, 2]]
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.
Original entry on oeis.org
-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
-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.
-
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]]
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.
Original entry on oeis.org
12, 64, 832, 576, 4060, 86428, 8800, 76000, 17500, 61500, 22243, 303810, 60333, 36672, 3045440, 42588, 114244, 48552, 1251081, 486387, 579734, 209409, 19615484, 281216, 10826816, 406848, 378211392, 43922220, 1051200, 1354560, 9939228, 66545721, 773916, 9585212
Offset: 1
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.
-
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, 2]]
Showing 1-3 of 3 results.
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