Sister Beiter conjecture

In mathematics, the Sister Beiter conjecture is a conjecture about the size of coefficients of ternary cyclotomic polynomials (i.e. where the index is the product of three prime numbers). It is named after Marion Beiter, a Catholic nun who first proposed it in 1968.

Background

For $n\in \mathbb {N} _{>0}$ the maximal coefficient (in absolute value) of the cyclotomic polynomial $\Phi _{n}(x)$ is denoted by $A(n)$.

Let $3\leq p\leq q\leq r$ be three prime numbers. In this case the cyclotomic polynomial $\Phi _{pqr}(x)$ is called ternary. In 1895, A. S. Bang proved that $A(pqr)\leq p-1$. This implies the existence of $M(p):=\max \limits _{p\leq q\leq r{\text{ prime}}}A(pqr)$ such that $1\leq M(p)\leq p-1$.

Statement

Sister Beiter conjectured in 1968 that $M(p)\leq {\frac {p+1}{2}}$. This was later disproved, but a corrected Sister Beiter conjecture was put forward as $M(p)\leq {\frac {2}{3}}p$.

Status

A preprint from 2023 explains the history in detail and claims to prove this corrected conjecture. Explicitly it claims to prove

$$M(p)\leq {\frac {2}{3}}p{\text{ and }}\lim \limits _{p\rightarrow \infty }{\frac {M(p)}{p}}={\frac {2}{3}}.$$