Sister Beiter conjecture explained

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 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} A(pqr) such that 1 \leq M(p)\leq p -1 .

Statement

Sister Beiter conjectured in 1968 that

M(p)\leq (p +1)/(2) . This was later disproved, but a corrected Sister Beiter conjecture was put forward as M(p)\leq (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$$M(p)\backslash leq\backslash fracp\; \backslash text\; \backslash lim\backslash limits\_\backslash frac=\; \backslash frac.$$