On the factorization of \(X^ n - BX - A\)
Generalizing a theorem of \textit{S. Rabinowitz} [Math. Mag. 61, 191-193 (1988; Zbl 0649.10010)] about polynomials \(x^ 5-x+A\), the author proves that if \(n\geq 5\) and either \(A\) or \(B\) is given then there are only finitely many polynomials \(x^ n-Bx-A\) with integer coefficients, divisible by a quadratic polynomial \(x^ 2-bx-a\) with integer coefficients. Moreover (via Thue theorem) these may be explicitly determined. The proof involves studying related recurrence sequences. As an example he notes that \(x^ 5+x+A\) has a quadratic factor for \(A=\pm 1\) or \(\pm 6\), and this corresponds to the fact that 1 and 144 are the only non-zero squares in the Fibonacci sequence.
divisibility by a quadratic polynomial with integer coefficients
scientific article; zbMATH DE number 55788