# A proof of the Corrected Beiter conjecture

## TL;DR

This paper proves the Corrected Beiter conjecture, establishing an upper bound on the maximum coefficient of cyclotomic polynomials of order three, which refines previous conjectures and corrects earlier false claims.

## Contribution

The paper provides a rigorous proof of the Corrected Beiter conjecture, confirming the new upper bound for coefficients of cyclotomic polynomials of order three.

## Key findings

- Confirmed the bound A(pqr) <= 2p/3 for cyclotomic polynomials of order three.
- Resolved a long-standing conjecture in algebraic number theory.
- Refined previous bounds and corrected earlier false conjectures.

## Abstract

We say that a cyclotomic polynomial \Phi_{n}(x) has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of \Phi_{n}(x) and M(p) be the maximum of A(pqr). In 1968, Sister Marion Beiter conjectured that A(pqr)<=(p+1)/2. In 2008, Yves Gallot and Pieter Moree showed that the conjecture is false for every p>=11, and they proposed the Corrected Beiter conjecture: A(pqr)<=2p/3. Here we will give a proof of this conjecture.