Power Products in Elliptic Divisibility Sequences and Prime-Incidence Obstructions

Let $E\mathrm{/}\mathbb{Q}$ be an elliptic curve, let $P\in E(\mathbb{Q})$ be non-torsion, and let $(D_n)$ be the associated elliptic divisibility sequence. For a fixed prime $\rho$, we study when an arbitrary finite product $$\prod _{i=1}^k{D}_{n_i}$$ can be a $\rho$-th power in ${\mathbb{Q}}^{\times }$. The main result is that, under the hypothesis that $D_1$ is divisible by $2$ or $3$, such product relations impose rigid restrictions on the large prime divisors of the indices $n_i$. More precisely, for every $B\ge 2$, all sufficiently large prime divisors $\mathrm{\ell }$ which occur as simple largest prime divisors of the indices and whose complementary cofactors are $B$-smooth must occur in $\rho$-balanced blocks. Equivalently, the corresponding prime-incidence rows over ${\mathbb{F}}_{\rho }$ have pairwise disjoint supports, are linearly independent, and satisfy the packing bound $$\mathrm{\mid }{\mathrm{\Lambda }}^{\ast }\mathrm{\mid }\le \lfloor k\mathrm{/}\rho \rfloor \mathrm{.}$$ In particular, if $n_i={\mathrm{\ell }}_i{a}_i$, where the ${\mathrm{\ell }}_i$ are sufficiently large primes and the $a_i$ are $B$-smooth, then a $\rho$-th power product relation can hold only if each prime $\mathrm{\ell }$ occurs among the ${\mathrm{\ell }}_i$ with multiplicity divisible by $\rho$. The proof combines Silverman’s valuation law, a fixed finite-prime-set consequence of Reynolds’ finiteness theorem, and the Hasse bound. The case $\rho =2$ gives the corresponding square-product obstruction.