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On Distinct Integers with Identical Prime Factor Sets for Three Consecutive Shifts

Naoki Takata
January 12, 2026

Abstract

We prove the existence of two distinct integers x and y such that the pairs (x, y), (x+1, y+1), and (x+2, y+2) each share the same set of prime factors. We provide an explicit construction using x = 2 and y = −4, and verify this result using the Lean 4 theorem prover with Mathlib.

Proof

PDF

Blockchain Timestamp (Proof of Existence)

As a proof of existence for this paper, the Keccak-256 hash of the PDF has been recorded on the Ethereum blockchain.

• Sha3-256: f7f9878add834fa7fdee8fa8fafa2802a0275f195e883b5b5955b8f77c258433
• Keccak-256: 41467fdafb35cb980554a8ab9edd6c2aa5765d608675a5878b61938bb1c57e64
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Timestamp: January 12, 2026 (date recorded on the blockchain)

Formal Verification

All theorems are formalized in Lean 4 (Mathlib commit f897ebcf72cd16f89ab4577d0c826cd14afaafc7). The verification used Aristotle, an automated theorem proving system for Lean 4.

Contact

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

This is a preprint. Comments are welcome!