Trinomials with integral \(S\)-unit coefficients having a quadratic factor (Q1678008)

Let \(S\) be a fixed finite set of primes. The authors provide finiteness results concerning trinomials of the shape \(x^n-Bx+A\) having a quadratic factor, where \(A,B\) are assumed to be integers having all their prime factors in \(S\). Beside a general ineffective finiteness theorem, they prove that all trinomials of the above shape having a quadratic factor different from \(x^2\pm x+1\), can be effectively determined. For the particular choice \(S=\{2,3,5,7\}\), more precise results are given. To prove their theorems, the authors combine several tools, including the primitive prime divisors of Lucas sequences and the theory of \(S\)-unit equations.