On a generalization of the Beiter Conjecture

Bartłomiej Bzdęga¹

¹ Faculty of Mathematics and Computer Sciences Adam Mickiewicz University 61-614 Poznań, Poland

Acta Arithmetica, Tome 173 (2016) no. 2, pp. 133-140

Résumé

We prove that for every $\varepsilon \gt 0$ and every nonnegative integer $w$ there exist primes $p_1,\ldots,p_w$ such that for $n=p_1\ldots p_w$ the height of the cyclotomic polynomial $\varPhi_n$ is at least $(1-\varepsilon)c_w M_n$, where $M_n=\prod_{i=1}^{w-2}p_i^{2^{w-1-i}-1}$ and $c_w$ is a constant depending only on $w$; furthermore $\lim_{w\to\infty}c_w^{2^{-w}}\approx0.71$. In our construction we can have $p_i \gt h(p_1\ldots p_{i-1})$ for all $i=1,\ldots,w$ and any function $h:\mathbb{R}_+\to\mathbb{R}_+$.

Détail

DOI : 10.4064/aa8119-1-2016

Keywords: prove every varepsilon every nonnegative integer there exist primes ldots ldots height cyclotomic polynomial varphi least varepsilon n where prod w w i constant depending only furthermore lim infty w approx construction have ldots i ldots function mathbb mathbb