Counting Proper Powers of Metallic Means in Arithmetic Progressions

The family of metallic means δₙ = (n + √(n² + 4))/2, which generalize the golden ratio, consists of quadratic units whose odd powers evolve as recurring metallic means, a discovery made by Lucas (1878). The associated trace polynomials are Dickson polynomials, whose value sets are classical. We study how the indices of these proper powers are distributed in arithmetic progressions. Möbius inversion gives an exact formula in every residue class. For fixed q and b, the leading term is ρₚ(q, b)·X^(1/p)/q, where p is the least odd prime whose trace polynomial represents b modulo q and ρₚ counts its seed residues. Such a prime always exists, and every odd prime occurs for a reduced class of some squarefree modulus, so the possible leading exponents are exactly {1/p : p an odd prime}. At squarefree moduli, admissibility is decided by a finite blocked set of primes determined by power-residue conditions. For a prime modulus, this gives the exact distribution of leading primes over the residue classes. Our main quantitative result concerns the smallest squarefree modulus Qₘᵢₙ(p) realizing exponent 1/p: a blocking argument and the Bombieri–Vinogradov theorem give p(1 + o(1)) ≤ log Qₘᵢₙ(p) ≤ (2 + o(1))p unconditionally, and log Qₘᵢₙ(p) ∼ p under the Elliott–Halberstam conjecture. Exact values for p ≤ 29 lie close to the lower bound.