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Large smooth twins from short lattice vectors
[Submitted on 22 Sep 2025 (v1), last revised 23 Jun 2026 (this v · 2026-06-24 · via math updates on arXiv.org

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Abstract:Finding the largest pair of consecutive $B$-smooth integers is computationally challenging. Current algorithms to find such pairs have an exponential runtime -- which has only be provably done for $B \leq 100$ and heuristically for $100 < B \leq 113$. We improve this by detailing a new algorithm to find such large pairs. The core idea is to solve the shortest vector problem (SVP) in a well constructed lattice. With this we are able to significantly increase $B$ and notably report the heuristically largest pair with $B = 751$ which has $196$-bits. By slightly modifying the lattice, we are able to find larger pairs for which one cannot conclusively say whether it is the largest or not for a given $B$. This notably includes a $213$-bit pair with $B = 997$ which is the largest pair found in this work.

Submission history

From: Benjamin Smith [view email] [via CCSD proxy]
[v1] Mon, 22 Sep 2025 12:34:56 UTC (4,209 KB)
[v2] Tue, 23 Jun 2026 08:02:14 UTC (2,796 KB)

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