The 3SUM problem asks whether a given collection of 
numbers contains three distinct
elements whose sum is zero. Its integer and real variants
are fundamental problems in the study of algorithms.
The 3SUM hypothesis asserts that no algorithm solves
the problem in time 
for any fixed 
.
Alman and Vassilevska Williams (2026) reported a deterministic algorithm with running time 
for integers of magnitude bounded by a polynomial
in 
, on a random
access machine with 
-bit
words. Their method also gives a truly subcubic all-pairs
shortest path algorithm. They state that known
reductions extend the refutation of the 3SUM hypothesis to the real-input version.
Alman and Vassilevska Williams (2026) state that Anthropic's Claude discovered the core algorithm, which they checked, simplified, and extended. They also report a Lean 4 formalization and take responsibility for the results.
