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 ∈ N > 0 {\displaystyle n\in \mathbb {N} _{>0}} the maximal coefficient (in absolute value) of the cyclotomic polynomial Φ n ( x ) {\displaystyle \Phi _{n}(x)} is denoted by A ( n ) {\displaystyle A(n)} . Let 3 ≤ p ≤ q ≤ r {\displaystyle 3\leq p\leq q\leq r} be three prime numbers. In this case the cyclotomic polynomial Φ p q r ( x ) {\displaystyle \Phi _{pqr}(x)} is called ternary. In 1895, A. S. Bang proved that A ( p q r ) ≤ p − 1 {\displaystyle A(pqr)\leq p-1} . This implies the existence of M ( p ) := max p ≤ q ≤ r prime A ( p q r ) {\displaystyle M(p):=\max \limits _{p\leq q\leq r{\text{ prime}}}A(pqr)} such that 1 ≤ M ( p ) ≤ p − 1 {\displaystyle 1\leq M(p)\leq p-1} .
Statement Sister Beiter conjectured in 1968 that M ( p ) ≤ p + 1 2 {\displaystyle M(p)\leq {\frac {p+1}{2}}} . This was later disproved, but a corrected Sister Beiter conjecture was put forward as M ( p ) ≤ 2 3 p {\displaystyle 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 ) ≤ 2 3 p and lim p → ∞ M ( p ) p = 2 3 . {\displaystyle M(p)\leq {\frac {2}{3}}p{\text{ and }}\lim \limits _{p\rightarrow \infty }{\frac {M(p)}{p}}={\frac {2}{3}}.}
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