A 55-Year-Old Math Problem Was Solved—This Time by Humans

Mathematicians have proved a conjecture posed by Ronald Graham in 1971: any set of distinct, non-zero numbers in arithmetic modulo a prime can be ordered so that its successive partial sums are all different. The result resolves a problem that resisted proof for 55 years and has a juggling analogy: ordering throws so that balls do not land on the same beat.

The solution came through four papers by several mathematicians, with the final paper by Lisa Sauermann and Huy Tuan Pham published in February 2026. The problem concerns clock arithmetic, where numbers repeat after a prime number p, and the difficulty varies with the size of the chosen set. The researchers used different approaches for different cases, with randomness playing a central role.