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 the University of California, Los Angeles, 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 long line of work on the mathematical impossibility of complete disorder. It also marks the first progress in decades on one of the biggest unsolved problems in the field of combinatorics.

“It’s phenomenally impressive that they managed to do this,” said Ben Green, a mathematician at the University of Oxford. At the time the work was released, the trio were all still in graduate school.

Sequences of regularly spaced numbers are called arithmetic progressions. Though they’re simple patterns, they hide astounding mathematical complexity. And they’re difficult, often impossible, to avoid, no matter how hard you might try.

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