A note on some properties of the $\lambda$-Polynomial

TL;DR

This paper investigates properties of the polynomial bla_n(a,b), derived from the factorization of a^n + b^n for odd n, revealing prime divisibility constraints under certain conditions.

Contribution

It proves specific properties of bla_n(a,b), including prime divisibility bounds when a and b are coprime and n is an odd prime.

Findings

Primes dividing bla_n(a,b) satisfy p  n for coprime a, b

The polynomial bla_n(a,b) has particular divisibility properties for odd n

Results extend understanding of polynomial factorization related to sum of powers

Abstract

The expression $a^n + b^n$ can be factored as $(a+b)(a^{n-1} - a^{n-2} b + a^{n-3} b^2 - ... + b^{n-1})$ when $n$ is an odd integer greater than one. This paper focuses on proving a few properties of the longer factor above, which we call $\lambda_n(a,b)$. One such property is that the primes which divide $\lambda_n(a,b)$ satsify $p \ge n$, if $a,b$ are coprime integers and $n$ is an odd prime.