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Finiteness of Powerful Numbers of the Form 2ⁿ ± 1 Under the ABC Conjecture

Naoki Takata
January 13, 2026

Abstract

We prove that, assuming the ABC conjecture, the set of natural numbers n such that 2ⁿ + 1 is a powerful number is finite, and similarly for 2ⁿ − 1. The proof relies on constructing ABC triples from powerful numbers and applying the radical bound implied by the ABC conjecture.

Proof

PDF

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Timestamp: January 13, 2026 (date recorded on the blockchain)

Contact

Naoki Takata
X: @naokitakata
Email: [ntakata [at] proton.me]

This is a preprint. Comments are welcome!