Valuation Separation for Coprime Lucas Products

Let $U_n=U_n(P,Q)$ be a nondegenerate Lucas sequence with $Q=\pm 1$ and discriminant $\mathrm{\Delta }=P^2+4Q>0$. We study Diophantine equations $$A{y}^k=\prod _{i=1}^r{U}_{n_i}(P,Q),k\ge 2,$$ where the indices $n_1,\dots ,n_r$ are pairwise coprime. The strong divisibility property implies that the factors $U_{n_i}$ are pairwise coprime, and hence a global $k$-th power condition separates into local valuation conditions on the individual factors. For $k=2$, this gives a termwise square-class restriction: each $U_{n_i}$ has signed squarefree part supported on the primes dividing $A$. In particular, the equation $\mathrm{\Delta }{y}^2=U_m{U}_n$, with $\gcd (m,n)=1$, reduces to a finite square-class compatibility condition together with an integrality condition. Assuming the number-field $abc$ conjecture over $\mathbb{Q}(\sqrt{\mathrm{\Delta }})$, we prove that only finitely many Lucas terms have squarefree part supported on a fixed finite set of rational primes. Consequently, the coprime product equations above admit an $abc$-conditional finite reduction. We also give the corresponding $k$-th power analogue and a primitive-divisor obstruction.