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Ternary cyclotomic polynomials having a large coefficient
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Yves Gallot
Published/Copyright:
June 16, 2009
Abstract
Let Φn(x) denote the nth cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that an(k), the coefficient of xk in Φn(x), satisfies |an(k)| ≦ (p + 1)/2 in case n = pqr with p < q < r primes (in this case Φn(x) is said to be ternary). Since then several results towards establishing her conjecture have been proved (for example |an(k)| ≦ 3p/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.
Received: 2007-12-12
Revised: 2008-03-19
Published Online: 2009-06-16
Published in Print: 2009-July
© Walter de Gruyter Berlin · New York 2009
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Articles in the same Issue
- Ends of locally symmetric spaces with maximal bottom spectrum
- Subsystems of finite type and semigroup invariants of subshifts
- Explicit n-descent on elliptic curves, II. Geometry
- On function fields with free absolute Galois groups
- Ternary cyclotomic polynomials having a large coefficient
- The Minus Conjecture revisited
- On the moduli space of certain smooth codimension-one foliations of the 5-sphere by complex surfaces
- Arithmetic duality theorems for 1-motives over function fields
- Arithmetic duality theorems for 1-motives
