Power Values of Certain Quadratic Polynomials

TL;DR

This paper investigates the power values of specific quadratic polynomials with negative squarefree discriminants, using algebraic number theory and primitive divisor theory to identify perfect powers in related integer sequences.

Contribution

It provides a method to compute power values of quadratic polynomials with certain discriminants and applies this to find perfect powers in polynomial-generated sequences.

Findings

Determined all perfect power terms in the Sylvester sequence.

Established bounds on the exponents for power values of the polynomials.

Connected polynomial power values to primitive divisor theory.

Abstract

In this article we compute the $q$th power values of the quadratic polynomials $f$ with negative squarefree discriminant such that $q$ is coprime to the class number of the splitting field of $f$ over $\mathbb{Q}$. The theory of unique factorisation and that of primitive divisors of integer sequences is used to deduce a bound on the values of $q$ which is small enough to allow the remaining cases to be easily checked. The results are used to determine all perfect power terms of certain polynomially generated integer sequences, including the Sylvester sequence.