Migotti's theorem on binary cyclotomic polynomials

For distinct primes p ≠ q, every coefficient of Φ_{pq} is -1, 0 or 1 (Migotti, 1883); for three primes it fails (Φ_{105} has a coefficient -2). Following Lam–Leung, with a p + b q = p q + 1 (1 ≤ a ≤ q, 1 ≤ b ≤ p, from Bézout), Φ_{pq} = A B - X C D where A, B, C, D are geometric sums in X^p and X^q; the identity is checked after multiplying by (X^p - 1)(X^q - 1), using (X^p - 1)(X^q - 1) Φ_{pq} = (X - 1)(X^{pq} - 1) from Mathlib's prod_cyclotomic_eq_X_pow_sub_one. The exponents ip + jq in each product are distinct, so its coefficients are 0 or 1, and the two supports are disjoint.

Main results

• Migotti.coeff_cyclotomic_mul_mem: coefficients of Phi_pq are -1, 0 or 1 for distinct primes p, q

References

• A. Migotti, Zur Theorie der Kreisteilungsgleichung, Sitzungsber. Akad. Wiss. Wien 87 (1883), 7–14.
• T. Y. Lam and K. H. Leung, On the cyclotomic polynomial Φ_pq(X), Amer. Math. Monthly 103 (1996), 562–564.

theorem Migotti.coeff_cyclotomic_mul_mem (p q : ℕ) (hp : Nat.Prime p) (hq : Nat.Prime q) (hpq : p ≠ q) (k : ℕ) : (Polynomial.cyclotomic (p * q) ℤ).coeff k ∈ {-1, 0, 1}

Migotti (1883). For distinct primes p ≠ q, every coefficient of the cyclotomic polynomial Φ_{pq} over ℤ is -1, 0 or 1. Proof: Lam–Leung (1996).