On a generalization of the Beiter Conjecture

Bartłomiej Bzdęga

Acta Arithmetica, Tome 172 (2016), p. 133-140 / Harvested from The Polish Digital Mathematics Library

Résumé

We prove that for every ε > 0 and every nonnegative integer w there exist primes $p_{1}, . . ., p_{w}$ such that for $n = p_{1} . . . p_{w}$ the height of the cyclotomic polynomial $Φ_{n}$ is at least $(1 - ε) c_{w} M_{n}$ , 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 $l i m_{w \to \infty} c_{w}^{2^{- w}} \approx 0.71$ . In our construction we can have $p_{i} > h (p_{1} . . . p_{i - 1})$ for all i = 1,...,w and any function h: ℝ₊ → ℝ₊.

Publié le : 2016-01-01

Zbl 06586878

EUDML-ID : urn:eudml:doc:279437

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     author = {Bart\l omiej Bzd\k ega},
     title = {On a generalization of the Beiter Conjecture},
     journal = {Acta Arithmetica},
     volume = {172},
     year = {2016},
     pages = {133-140},
     zbl = {06586878},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8119-1-2016}
}

Bartłomiej Bzdęga. On a generalization of the Beiter Conjecture. Acta Arithmetica, Tome 172 (2016) pp. 133-140. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8119-1-2016/