On the Flatness of Ternary Cyclotomic Polynomials

A cyclotomic polynomial Φ _n ( x ) is said to be ternary if n = p ₁ p ₂ p ₃ , where p ₁ < p ₂ < p ₃ are three distinct odd primes. In this note, we determine a set of explicit exponents in the cyclotomic polynomial , where p ₂ ≡ ±2(mod p ₁ ) and p ₃ ≡ 2(mod p ₁ p ₂ ), whose coefficients are equal to 2. Moreover, for , where p ₂ ≡ −1(mod p ₁ ) and p ₃ ≡ 2(mod p ₁ p ₂ ), we determine all exponents whose coefficients are equal to 2. Additionally, for , where p ₂ ≡ 2, 3(mod5) and p ₃ ≡ 2(mod5 p ₂ ), we determine all exponents whose coefficients are equal to 2.