On the coefficients of a class of ternary cyclotomic polynomials

A cyclotomic polynomial \(\Phi_n(x)\) is said to be flat if its nonzero coefficients involve only \(\pm1\). In this paper, for odd primes \(p \lt q \lt r\) with \(q\equiv 1\pmod p\) and \(9r\equiv \pm1\pmod {pq}\), we prove that \(\Phi_{pqr}(x)\) is flat if and only if \(p=5\), \(q\geq 41\), and \(q\equiv 1\pmod 5\).