Large smooth twins from short lattice vectors

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.