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- Title
Ternary cyclotomic polynomials having a large coefficient.
- Authors
Gallot, Yves; Moree, Pieter
- Abstract
Let Φ n( x) denote the nth cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that an( k), the coefficient of xk in Φ n( x), satisfies | an( k)| ≦ ( p + 1)/2 in case n = pqr with p < q < r primes (in this case Φ n( x) is said to be ternary). Since then several results towards establishing her conjecture have been proved (for example | an( k)| ≦ 3 p/4). Here we show that, nevertheless, Beiter's conjecture is false for every p ≧ 11. We also prove that given any ε > 0 there exist infinitely many triples ( pj, qj, rj) with p1 < p2 < ⋯ consecutive primes such that | apjqjrj( nj)| > (2/3 – ε) pj for j ≧ 1.
- Subjects
POLYNOMIALS; CYCLOTOMY; BEITER, Marion; PRIME numbers; ALGEBRA; LOGICAL prediction
- Publication
Journal für die Reine und Angewandte Mathematik, 2009, Vol 2009, Issue 632, p105
- ISSN
0075-4102
- Publication type
Article
- DOI
10.1515/CRELLE.2009.052