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 $n\in \mathbb {N} _{>0}$ 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^[2] 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^[1] 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^[3] from 2023 explains the history in detail and claims to prove this corrected conjecture. Explicitly it claims to prove $M(p)\leq {\frac {2}{3}}p{\text{ and }}\lim \limits _{p\rightarrow \infty }{\frac {M(p)}{p}}={\frac {2}{3}}.$

References

^ ^a ^b Beiter, Marion (April 1968). "Magnitude of the Coefficients of the Cyclotomic Polynomial $F_{pqr}(x)$ ". The American Mathematical Monthly. 75 (4): 370–372. doi:10.2307/2313416. JSTOR 2313416.
^ Bang, A.S. (1895). "Om Ligningen $\Phi _{n}(x)=0$ ". Tidsskr. Math. 6: 6–12.
^ 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].

[paper68-1] Beiter, Marion (April 1968). "Magnitude of the Coefficients of the Cyclotomic Polynomial $F_{pqr}(x)$ ". The American Mathematical Monthly. 75 (4): 370–372. doi:10.2307/2313416. JSTOR 2313416.

[bang-2] Bang, A.S. (1895). "Om Ligningen $\Phi _{n}(x)=0$ ". Tidsskr. Math. 6: 6–12.

[arxiv23-3] 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].

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