Phạm Tuấn Huy, a mathematics professor at the University of Chicago, has collaborated with University of Bonn professor Lisa Sauermann to resolve a 55-year-old mathematical problem known as Graham’s rearrangement conjecture. The duo published their 27-page proof on arXiv in February 2026.
The puzzle dates back to 1971, when mathematician Ronald Graham posed a deceptively simple question involving finite sets of distinct, non-zero integers. Graham, who served as president of the American Mathematical Society and was also an accomplished juggler and president of the International Jugglers’ Association, is believed by some mathematicians to have drawn inspiration for the problem from the mechanics of juggling. The core question asks whether a set of distinct integers can always be arranged such that when taking the sum of the first two numbers, then the first three, then the first four, and so on, every resulting partial sum remains distinct.
In terms of physical performance, the dilemma translates to a juggling scenario where each object stays in the air for a distinct amount of duration. The challenge asks whether a juggler can always find an arrangement where no two objects drop at the exact same beat to ruin the performance. While positive answers are trivial for sets of positive numbers where sums only grow, the hurdle emerges in finite worlds where numbers wrap around like clock faces, creating overlapping sums.
Resolving the Midsize Gap Left by Decades of Partial Proofs
That gap closed after Phạm Tuấn Huy and Sauermann attended a conference in Germany in September 2025. Listening to two presentations on the conjecture pulled them into the problem. Huy spent an extra three days in Bonn, and by the end of the visit, the pair mapped out a strategy to clear the remaining void.
Over the decades, mathematicians chipped away at Graham’s conjecture in distinct pieces. Alp Müyesser and Alexey Pokrovskiy tackled very large sets starting in 2022. Noah Kravitz and Benjamin Bedert addressed very small sets in 2024, with colleagues further expanding those boundaries in August 2025. Yet a stubborn middle ground remained untouched, leaving midsize integer sets completely unhandled by legacy methods.
According to Müyesser, the duo utilized an approach completely distinct from previous attempts. Rather than sticking to standard tactics, the researchers randomly shuffled the integer set and designed an error-correction procedure. Whenever a segment added up to zero, they replaced the final number of that segment.
Applying Anti-Concentration and Fourier Analysis to Complex Mechanics
Proving that their correction procedure almost always succeeds required working through three distinct ways the system could fail. To achieve this, Huy and Sauermann deployed a notoriously intricate technique known as anti-concentration, alongside Fourier analysis—a mathematical framework that breaks functions down into sums of simple waves.
Their analysis demonstrated that when random number sets are added together, no single sum appears with an overwhelmingly high probability. Consequently, they proved that the combined probability of all three failure modes occurring stays under 100 percent, confirming that Graham’s desired ordering always exists.
The methodology proved so formidable that Kravitz admitted his own research group lacked the courage to test this direction. Despite its sweeping success across arbitrary set sizes, the proof carries practical limits, applying strictly to sufficiently large prime numbers scaling near 10 to the power of 100—meaning a literal juggling performance matching the math would run for an impossible duration.
Tracing an Accelerated Academic Trajectory at Stanford, Chicago, and Princeton
Phạm Tuấn Huy's path to resolving the 55-year-old riddle spans elite institutions across continents. He then moved to the United States to study at Stanford University, where he earned a bachelor's degree in mathematics with honors, a minor in computer science, and a master's degree in statistics.
During his undergraduate years, Huy ranked four years among the top 80 scorers in the Putnam competition, widely considered North America’s hardest collegiate math contest. His Stanford undergraduate thesis won the Kennedy Prize for best natural sciences thesis in 2018. He subsequently earned a master’s degree with distinction from the University of Cambridge, ranking first in the Part III Tripos examinations in 2019.
Huy completed his Ph.D. at Stanford in 2023 under adviser Jacob Fox. That same year, the Clay Mathematics Institute appointed him as a research fellow for a five-year term running from July 2023 through 2028, making him the second Vietnamese mathematician to receive the honor after Ngô Bảo Châu in 2004. His previous major milestones include proving the Kahn-Kalai conjecture alongside Jinyoung Park, published in the Journal of the American Mathematical Society in 2024.
Now working at the University of Chicago, Huy has attained the rank of full professor—an unusual milestone in the United States academic system, where faculty members typically require 10 to 20 years to reach the top tier. Caltech records show he was appointed as an assistant professor in 2025 following a two-year visiting stint, and he transitions to the Institute for Advanced Study in Princeton. His co-author, Sauermann, is a 34-year-old professor at the University of Bonn with four IMO gold medals of her own, with the two mathematicians having known each other for a decade since their time together at Stanford.