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A371553 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 A371553 #16 Apr 20 2024 16:35:55
%S A371553 11,15,15,20,120,120,280,280,312,330,330,750,750,4095,4095,5700,5700,
%T A371553 7800,7800,10140,10140,10564,10564,11102,11275,11275,21970,21970,
%U A371553 27248,30758,31000,31000,31146,31350,31350,32955,32955,35490,35490,38360,38360,41236
%N A371553 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.
%C A371553 Define the equivalence relation ~ on pairs of nonzero rational numbers by (b, c) ~ (b', c') if there exists a nonzero rational number k such that b' = k^4*b and c' = k^5*c. Every such pair (b, c) is equivalent to a unique pair of integers (b', c') with c' > 0 and |b'| as small as possible, which we call a primitive pair. If (b, c) ~ (b', c') and x^5 + b*x + c is irreducible and solvable by radicals, then so is x^5 + b'*x + c' by making the substitution x -> x/k and multiplying by k^5. Hence, every polynomial of the form x^5 + b*x + c with b, c nonzero rationals is equivalent to one with integer coefficients and positive constant coefficient.
%C A371553 An irreducible polynomial of the form x^5 + b*x + c for rational b, c is solvable by radicals if and only if its Galois group is a subgroup of the Frobenius group of order 20, which happens if and only if the resolvent sextic (x - b)^4*(x^2 - 6*b*x + 25*b^2) - 3125*c^4*x has a rational root. If b and c are integers, then such a rational root x must be an integer, by the rational root theorem. Therefore, given an integer b, we can find all such integers c by solving the quadratic Diophantine equation x^2 - (6*b + 5*y)*x + 25*b^2 = 0 for x and y, which has finitely many solutions. The values of c are then a subset of the values +-(x-b)*y^(1/4)/5.
%H A371553 Ben Whitmore, <a href="/A371553/b371553.txt">Table of n, a(n) for n = 1..95</a>
%F A371553 x^5 + a(n)*x + A371554(n) is irreducible and solvable by radicals.
%e A371553 15 is in the sequence twice because x^5 + 15*x + 12 and x^5 + 15*x + 44 are both irreducible and solvable by radicals, and (15, 12) and (15, 44) are both primitive pairs.
%e A371553 176 is not in the sequence because there is no integer c for which (176, c) is primitive and x^5 + 176*x + c is irreducible and solvable by radicals. x^5 + 176*x + 1408 is irreducible and solvable by radicals, but (176, 1408) is not primitive because it is equivalent to (11, 44).
%e A371553 x^5 + (10/13)*x - 3/13 is solvable by radicals, and (10/13, -3/13) ~ (21970, 85683) which is primitive, so 21970 is in the sequence.
%t A371553 pairs = Join @@ Table[
%t A371553   Select[{b, Abs[#1 - b] #2/5} & @@@
%t A371553     Sort[SolveValues[x^2 - (6b + 5y^4)x + 25b^2 == 0 && y > 0, {x, y}, Integers]],
%t A371553     Max[Last /@ FactorInteger[GCD @@ #]] < 4 &&
%t A371553     AllTrue[#, IntegerQ] &&
%t A371553     IrreduciblePolynomialQ[x^5 + #1x + #2 & @@ #] &
%t A371553   ],
%t A371553   {b, 1, 1000}
%t A371553 ];
%t A371553 pairs[[All, 1]]
%Y A371553 For values of c see A371554.
%K A371553 nonn
%O A371553 1,1
%A A371553 _Ben Whitmore_, Mar 27 2024