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arXiv · 2509.17699

Large smooth twins from short lattice vectors

Erik Mulder·Bruno Sterner·Wessel van Woerden
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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.

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Erik Mulder, Bruno Sterner, Wessel van Woerden. 2025-09-22. Large smooth twins from short lattice vectors. https://arxiv.org/abs/2509.17699

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