Power Products in Elliptic Divisibility Sequences and Prime-Incidence Obstructions

Abstract

Let $E/\mathbb Q$ 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 $\ell$ 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 \[ |\Lambda^*|\le \lfloor k/\rho\rfloor . \] In particular, if $n_i=\ell_i a_i$, where the $\ell_i$ are sufficiently large primes and the $a_i$ are $B$-smooth, then a $\rho$-th power product relation can hold only if each prime $\ell$ occurs among the $\ell_i$ 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.