The conjecture lives in the realm of modular, or “clock,” arithmetic. Imagine placing the integers on a number line and then wrapping that line around a clock whose face repeats every prime number p. With p = 7, for example, the numbers 0, 7, 14, … are all equivalent, and adding two positive numbers such as 3 and 4 can even give zero in this setting.
Ron Graham posed a simple yet elusive question: If you pick any non‑zero residues modulo p, can you always rearrange them so that every partial sum you obtain along the way is distinct? The answer depends on how many residues you choose relative to the size of the modulus.
When the chosen set is huge—containing almost every possible residue modulo p—there are far too many ways the numbers can be ordered, making a valid arrangement difficult to construct. In 2022, Büyükmutlu (along with his former adviser Alexey Pokrovskiy of University College London) showed that starting from a random ordering and then repairing any “bad” zero‑sum subsequences could get you almost all the way there. Their method involved setting aside a few special numbers, randomly permuting the rest, and, whenever a zero‑sum interval appeared, swapping its final element with a spare number.
“It’s embarrassing for humanity that we don’t know this. This situation just had to be rectified,” said Oxford’s Noah Kravitz, echoing a sentiment that had guided his own work.
“Computer scientists often call this a ‘finding the hay in the haystack’ problem,” Müyesser explained. “You might know that lots of good orderings are out there, but actually finding one is hard. If you do it randomly, it’s likely going to work, but it’s hard to explicitly describe what the solution is supposed to look like.”
Kravitz and his collaborator Benjamin Bedert tackled the opposite extreme—when the set is tiny compared to p, say 100 numbers while p is a billion. Their random‑repair strategy also succeeded, and they posted a complete proof in September 2024.
Noah Kravitz is one of several young mathematicians who recently revived the decades‑old Graham conjecture.
Word of Kravitz and Bedert’s solution reached Müyesser, who shared his own large‑set work; together with Pokrovskiy, Kravitz, and Bedert (and two other colleagues) they produced papers covering many remaining cases. However, a “medium‑size” gap—sets containing roughly half as many residues as p—remained unsolved. “Our methods didn’t work there, and there were clear reasons that they would not have worked,” Müyesser noted.
It looked as though the problem would once again lapse into limbo, but a surprise emerged in February 2026.
The Fountain
Lisa Sauermann and Huy Tuan Pham had been friends since meeting at Stanford in 2015. A conference in Germany in September 2025 gave them a rare chance to share a chalkboard again, and Pham followed Sauermann to Bonn for a short visit. All they needed was a problem to work on.
At the conference they heard two talks on Graham’s conjecture by mathematicians who had attempted but failed to bridge the medium‑size gap. Sauermann, who as a high‑school student had solved a closely related problem on the International Mathematical Olympiad (a problem most likely placed by Erdős‑Chung), was intrigued. By the end of their three‑day visit, the pair had a plan that relied on a technically demanding method called anti‑concentration.
Lisa Sauermann (top) and Huy Tuan Pham have been friends for the past decade. Their recent collaboration on the Graham conjecture finally settled the problem. Barbara Frommann/University of Bonn; Courtesy of Huy Tuan Pham
Anti‑concentration asserts that certain events are especially unlikely. Sauermann and Pham first attempted the same random‑repair procedure that had worked for the large‑ and small‑set cases, but they identified three “bad events” that could thwart it: a zero‑sum sequence occurring at the very end of the ordering (leaving no spare numbers to swap), many zero‑sum sequences appearing too close together (making simultaneous fixes impossible), and fixing one bad sequence creating another downstream.
Using Fourier analysis, they showed that in a genuinely random sum of residues, no single sum dominates. This insight allowed them to bound the probability of each bad event and prove that the total chance of any bad event occurring was less than 100 %. Consequently, a suitable rearrangement always exists.
A few months later the duo posted a 27‑page proof online. Not only did they settle the medium‑size case, but they also demonstrated that a random ordering could be corrected to eliminate bad events at least 90 % of the time—an exceptionally high success rate.
“Their approach is just completely different,” Müyesser said when he learned of the result.
Together, the four papers now prove Graham’s conjecture for sets of all sizes—though they assume an astronomically large prime p (on the order of 10¹⁰⁰). The exact value of p remains unknown, and using the result for a real‑world juggling routine would require an impractically long performance.
The achievement confirms that even within these constrained modular worlds, “there are some nice structures that always exist,” remarked mathematician Nati Alon. “You can always achieve some degree of flexibility, shuffling the numbers in your set around to avoid revisiting the same partial sums.”
“To pose a good problem is really an art,” said Fan Chung. “I think Ron would be extremely happy to see the problem solved.”
