A proof of the Corrected Beiter conjecture

Jia Zhao, Xianke Zhang

Published 2009-10-15Version 1

We say that a cyclotomic polynomial \Phi_{n}(x) has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of \Phi_{n}(x) and M(p) be the maximum of A(pqr). In 1968, Sister Marion Beiter conjectured that A(pqr)<=(p+1)/2. In 2008, Yves Gallot and Pieter Moree showed that the conjecture is false for every p>=11, and they proposed the Corrected Beiter conjecture: A(pqr)<=2p/3. Here we will give a proof of this conjecture.