cp's OEIS Frontend

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.

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.