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 [math]\displaystyle{ n\in\mathbb{N}_{\gt 0} }[/math] the maximal coefficient (in absolute value) of the cyclotomic polynomial [math]\displaystyle{ \Phi_n(x) }[/math] is denoted by [math]\displaystyle{ A(n) }[/math].
Let [math]\displaystyle{ 3\leq p\leq q\leq r }[/math] be three prime numbers. In this case the cyclotomic polynomial [math]\displaystyle{ \Phi_{pqr}(x) }[/math] is called ternary. In 1895, A. S. Bang[2] proved that [math]\displaystyle{ A(pqr)\leq p-1 }[/math]. This implies the existence of [math]\displaystyle{ M(p):=\max\limits_{p\leq q\leq r\text{ prime}}A(pqr) }[/math] such that [math]\displaystyle{ 1\leq M(p)\leq p-1 }[/math].
Statement
Sister Beiter conjectured[1] in 1968 that [math]\displaystyle{ M(p)\leq \frac{p+1}{2} }[/math]. This was later disproved, but a corrected Sister Beiter conjecture was put forward as [math]\displaystyle{ M(p)\leq \frac{2}{3}p }[/math].
Status
A preprint[3] from 2023 explains the history in detail and claims to prove this corrected conjecture. Explicitly it claims to prove [math]\displaystyle{ M(p)\leq\frac{2}{3}p \text{ and } \lim\limits_{p\rightarrow\infty}\frac{M(p)}{p}= \frac{2}{3}. }[/math]
References
- ↑ 1.0 1.1 Beiter, Marion (April 1968). "Magnitude of the Coefficients of the Cyclotomic Polynomial [math]\displaystyle{ F_{pqr}(x) }[/math]". The American Mathematical Monthly 75 (4): 370–372. doi:10.2307/2313416.
- ↑ Bang, A.S. (1895). "Om Ligningen [math]\displaystyle{ \Phi_n(x)=0 }[/math]". 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].
 |  |
