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. We study when a product \[ \prod_{i=1}^k D_{n_i} \] can be a $\rho$-th power, where $\rho$ is a fixed prime. Our obstructions concern the incidence of prime divisors among the indices $n_i$, rather than the prime factorisation of individual EDS terms. Assuming that $D_1$ is divisible by $2$ or $3$, we show that sufficiently large primes $\ell$ occurring as simple largest prime divisors of the indices, with $n_i/\ell$ $B$-smooth whenever $\ell\mid n_i$, must occur in $\rho$-balanced incidence blocks. In particular, for a finite family of such primes, the associated incidence rows are disjoint, linearly independent over $\mathbb F_\rho$, and satisfy \[ |\Lambda^\ast|\le \lfloor k/\rho\rfloor . \]
| Comments: | 22 pages |
| Subjects: | Number Theory (math.NT) |
| MSC classes: | 11G05 (Primary) 11B37, 11D45, 11D61 (Secondary) |
| Cite as: | arXiv:2605.25797 [math.NT] |
| (or arXiv:2605.25797v1 [math.NT] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25797 arXiv-issued DOI via DataCite (pending registration) |
From: Dongyeon Kym [view email]
[v1]
Mon, 25 May 2026 12:48:16 UTC (13 KB)
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