### The family of ternary cyclotomic polynomials with one free prime

Yves Gallot, Pieter Moree, Robert Wilms
2011 Involve. A Journal of Mathematics
A cyclotomic polynomial n (x) is said to be ternary if n = pqr , with p, q and r distinct odd primes. Ternary cyclotomic polynomials are the simplest ones for which the behavior of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family pqr (x) with p < q < r , p and q fixed and r a free prime. MSC2000: primary 11C08; secondary 11B83. Keywords: ternary cyclotomic polynomial,
more » ... ient. 317 318 YVES GALLOT, PIETER MOREE AND ROBERT WILMS where 2 < p < q are fixed primes and r is a "free prime". Up to now in the literature the above family was considered, but with also q free. The maximum coefficient (in absolute value) that occurs in that family will be denoted by M( p), thus M( p) = max{A( pqr ) : p < q < r }, with p > 2 fixed. Similarly we define M( p; q) to be the maximum coefficient (in absolute value) that occurs in the family (1), thus M( p; q) = max{A( pqr ) : r > q}, with 2 < p < q fixed primes. Example. Bang [1895] proved that M( p) ≤ p − 1. Since a 3·5·7 (7) = −2 we infer that M(3) = 2. Using a 105 (7) = −2 and M(3) = 2, we infer that M(3; 5) = 2.