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.[1]

Background

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

Let ${\displaystyle 3\leq p\leq q\leq r}$ be three prime numbers. In this case the cyclotomic polynomial ${\displaystyle \Phi _{pqr}(x)}$ is called ternary. In 1895, A. S. Bang[2] proved that ${\displaystyle A(pqr)\leq p-1}$. This implies the existence of ${\displaystyle M(p):=\max \limits _{p\leq q\leq r{\text{ prime}}}A(pqr)}$ such that ${\displaystyle 1\leq M(p)\leq p-1}$.

Statement

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

Status

A preprint[3] from 2023 explains the history in detail and claims to prove this corrected conjecture. Explicitly it claims to prove $${\displaystyle M(p)\leq {\frac {2}{3}}p{\text{ and }}\lim \limits _{p\rightarrow \infty }{\frac {M(p)}{p}}={\frac {2}{3}}.}$$

References

1. Beiter, Marion (April 1968). "Magnitude of the Coefficients of the Cyclotomic Polynomial ${\displaystyle F_{pqr}(x)}$". The American Mathematical Monthly. 75 (4): 370–372. doi:10.2307/2313416. JSTOR 2313416.
2. Bang, A.S. (1895). "Om Ligningen ${\displaystyle \Phi _{n}(x)=0}$". Tidsskr. Math. 6: 6–12.
3. Juran, Branko; Moree, Pieter; Riekert, Adrian; Schmitz, David; Völlmecke, Julian (2023). "A proof of the corrected Sister Beiter cyclotomic coefficient conjecture inspired by Zhao and Zhang". arXiv:2304.09250 [math.NT].