Integer Sequences of the Form a^n + b^n

TL;DR

This paper characterizes all integer sequences of the form a^n + b^n with complex numbers a, b, linking them to quadratic roots and classical sequences like Lucas and Fibonacci, while also proving the non-existence of sequences of the form a^n - b^n.

Contribution

It establishes a correspondence between quadratic roots and integer sequences of the form a^n + b^n, and explores their connections to well-known number sequences.

Findings

Sequences of the form a^n + b^n are linked to roots of quadratic equations.

No integer sequences of the form a^n - b^n exist.

Formulas involving Lucas and Fibonacci numbers are derived.

Abstract

In this paper, we find all integer sequences of the form a^n + b^n, where a and b are complex numbers and n is a nonnegative integer. We prove that if p and q are integers, then there is a correspondence between the roots of the quadratic equation z^2 - pz - q = 0 and integer sequences of the form a^n + b^n. In addition, we will show that there are no integer sequences of the form a^n - b^n. Finally, we use special values of a and b to obtain a range of formulas involving Lucas and Fibonacci numbers.