The original version of this story appeared in Quanta Magazine.
In late 2017, Ashwin Sah and Mehtaab Sawhney met as undergraduates at the Massachusetts Institute of Technology. Since then, the pair have written a mind-boggling 57 math proofs together, many of them profound advances in various fields.
In February, Sah and Sawhney announced yet another joint accomplishment. With James Leng, a graduate student at UCLA, they obtained a long-sought improvement on an estimate of how big sets of integers can get before they must contain sequences of evenly spaced numbers, like {9, 19, 29, 39, 49} or {30, 60, 90, 120}. The proof joins a