Breakthroughs Weekly — 4 October 2026
A 1971 conjecture of Ronald Graham on reordering residues so all partial sums differ is resolved for large primes, now drawing wide expert attention.
Graham’s 1971 rearrangement conjecture is settled for all large primes
arXiv · 17 Feb 2026 · Combinatorics
Huy Tuan Pham and Lisa Sauermann proved that, for any prime p, a set of nonzero residues can always be listed in an order where every running total is different, a question Ronald Graham posed in 1971. Their paper handles sets up to a power-law fraction of the prime’s size, and combined with earlier work by Müyesser, Pokrovskiy, Kravitz, Bedert and colleagues, this settles the conjecture for all sufficiently large primes. The argument leans on randomness together with anti-concentration estimates and Fourier analysis. It was posted in February and is a preprint without a journal reference, but it reached wider attention on 28 September 2026 when Quanta Magazine profiled the full line of work, with Princeton’s Noga Alon among the experts commenting on it.