NonBeiter ternary cyclotomic polynomials with an optimally large set of coefficients
Abstract
Let l>=1 be an arbitrary odd integer and p,q and r primes. We show that there exist infinitely many ternary cyclotomic polynomials \Phi_{pqr}(x) with l^2+3l+5<= p<q<r such that the set of coefficients of each of them consists of the p integers in the interval [(pl2)/2,(p+l+2)/2]. It is known that no larger coefficient range is possible. The Beiter conjecture states that the cyclotomic coefficients a_{pqr}(k) of \Phi_{pqr} satisfy a_{pqr}(k)<= (p+1)/2 and thus the above family contradicts the Beiter conjecture. The two already known families of ternary cyclotomic polynomials with an optimally large set of coefficients (found by G. Bachman) satisfy the Beiter conjecture.
 Publication:

arXiv eprints
 Pub Date:
 November 2011
 arXiv:
 arXiv:1111.6800
 Bibcode:
 2011arXiv1111.6800M
 Keywords:

 Mathematics  Number Theory;
 11N37;
 11B83
 EPrint:
 20 pages, 7 Tables