Valuation Separation for Coprime Lucas Products

Abstract

Let $U_n=U_n(P,Q)$ be a nondegenerate Lucas sequence with $Q=\pm 1$ and discriminant $\Delta=P^2+4Q>0$. We study Diophantine equations \[ A y^k=\prod_{i=1}^r U_{n_i}(P,Q), \qquad k\geq 2, \] where the indices $n_1,\ldots,n_r$ 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 $\Delta y^2=U_mU_n$, 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 $\mathbb Q(\sqrt{\Delta})$, 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.