Large smooth twins from short lattice vectors
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 ≤ 100 and heuristically for 100 < B ≤ 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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BibTeX
@misc{mulder2026large,
title = {{Large smooth twins from short lattice vectors}},
author = {Erik Mulder and Bruno Sterner and Wessel van Woerden},
howpublished = {HAL},
year = {2026},
month = sep,
eprint = {2509.17699},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05207824v3},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Large smooth twins from short lattice vectors
AU - Erik Mulder
AU - Bruno Sterner
AU - Wessel van Woerden
PY - 2026
DA - 2026/09/02
PB - HAL
UR - https://hal.science/hal-05207824v3
AB - 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 ≤ 100 and heuristically for 100 < B ≤ 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.
ER -