MANY MINDEDMathematicians resolve modular arithmetic conjecture with probabilistic proof
Lisa Sauermann of the University of Bonn and Huy Tuan Pham of the University of Chicago resolved Ronald Graham’s 1971 rearrangement conjecture in four papers published February 2026. The conjecture asks whether nonzero integers modulo a prime p can be rearranged so all partial sums remain distinct—a question tied to juggling patterns in Graham’s original work. Their proof uses probabilistic methods to construct valid orderings within clock arithmetic, avoiding exhaustive checks that would be computationally infeasible for large sets.
This theoretical advance operates within strict constraints: it applies only to prime modulus p, not composite numbers. Earlier work by Alp Müyesser and Alexey Pokrovskiy in 2022 addressed a related but distinct case for large sets, meaning the 2026 resolution is the final step for this specific conjecture.
For computing and cryptography, this could enable more efficient algorithms that reduce computational overhead in systems requiring secure data handling. While the direct impact on cost remains theoretical, it strengthens the foundation for future advancements where modular arithmetic is used—potentially lowering barriers to secure, low-cost digital services. The real-world translation depends on how these methods scale to practical applications.
What to watch: Implementation in cryptographic protocols and whether probabilistic approaches can be adapted to real-world systems without compromising security. Caveats: The solution is confined to prime modulus, and earlier work addressed a different mathematical scenario.
Source: Quanta