Divisibility Properties and Sieve-Theoretic Analysis of a Parametric Family of Dickson-Type Polynomials

For each positive integer m, let β(m) = (1+√(1+4m))/2 be the positive root of x²−x−m = 0. For each odd prime P, define f_P(m) = β(m)^P − (β(m)−1)^P. This is a polynomial in m of degree (P−1)/2, related to — but distinct from — classical Dickson polynomials and Lucas sequences, with f_P(1) = L_P (Lucas numbers) and f_P(2) = 2^P − 1 (Mersenne numbers). We prove unconditionally: (i) f_P(m) ≡ 1 (mod P) for all m (Frobenius congruence); (ii) for any prime q < P, the polynomial f_P has no roots modulo q (uniform non-divisibility); and (iii) the overlap between any two members f_{P₁} and f_{P₂} is finite (by reduction to genus-≥1 curves and Siegel/Faltings), with complete disjointness for prime inputs. These divisibility properties imply that the sifting density of f_5(m) = 5m²+5m+1 unconditionally exceeds that of the benchmark polynomial n²+1 by a factor of 5/3·c₀ ≈ 1.615. We verify that f_5 satisfies the hypotheses of Iwaniec's half-dimensional sieve with the exception of explicit Type II bilinear estimates, which we outline but do not complete; under these hypotheses, 5m²+5m+1 would represent infinitely many P₂ numbers. Heuristic Bateman–Horn predictions are discussed and are conjectural, not affecting the unconditional results. A combined immunity theorem shows that seven family members collectively have no common prime divisor below 500.