Graham's 1971 rearrangement conjecture on distinct partial sums proved in four papers by 2026

A conjecture posed in 1971 by the mathematician Ronald Graham has been resolved in a proof spread over four papers, wired.com reports, citing Quanta Magazine. The final paper, by Lisa Sauermann of the University of Bonn and Huy Tuan Pham of the University of Chicago, appeared in February 2026 and officially closed the problem wired.comwired.comwired.comOpen source ↗.

Graham asked whether any set of distinct nonzero integers can be reordered so that all partial sums (first two numbers, first three, and so on) are different. For positive numbers, and for mixes of positive and negative numbers, the answer was known to be yes. The open case was arithmetic modulo a prime p, where numbers wrap around like a clock face and multiples of p are equivalent to zero.

Large and small sets

The difficulty depends on the size of the set relative to p. Alp Müyesser of the University of Oxford and Alexey Pokrovskiy of University College London handled sets that include almost every possible number up to p. They began with a random ordering and set aside a few specially chosen numbers. When a stretch of the ordering added up to zero, which would repeat a partial sum, they inserted a spare number to break it. They posted this in 2022, within a paper on a more general problem.

Noah Kravitz and Benjamin Bedert of Oxford approached from the other end: sets that are tiny compared with p, for example 100 numbers when p is 1 billion. They posted their proof in September 2024 wired.comwired.comwired.comOpen source ↗.

Closing the gap

Kravitz and Bedert then joined Müyesser and two other colleagues to extend the earlier approach. Their paper, posted in August 2025, covered more cases where the set is relatively large compared with p. A gap remained for medium-size sets, such as those with roughly half as many numbers as p, where, according to Müyesser, the existing methods did not work.

Sauermann and Pham met in 2015 at Stanford University. In September 2025 they attended a conference in Germany, where they heard two talks on Graham's conjecture by mathematicians who had tried and failed to prove it. Pham then visited Sauermann in Bonn. Their paper followed in February 2026.

Graham was also president of the American Mathematical Society and of the International Jugglers' Association. Some mathematicians now say the question may have been inspired by juggling: if each ball stays in the air for a different time, can the throws be ordered so that no two balls land on the same beat wired.comwired.comwired.comOpen source ↗.