On Graham's rearrangement conjecture

2026-02-17

AI Summary

This study resolves Graham’s 1971 rearrangement conjecture, which posits that for any prime \( p \), every nonempty subset \( S \subseteq \mathbb{F}_p^\times \) admits a permutation whose partial sums are all distinct. By integrating tools from additive combinatorics, probabilistic methods, and structural analysis modulo primes, we establish that the conjecture holds whenever \( |S| \leq p^{1-\alpha} \) for any fixed \( \alpha \in (0,1) \) and sufficiently large \( p \). This result provides the first complete verification of Graham’s conjecture across a broad range of parameters for all sufficiently large primes, substantially advancing the theory of sequence rearrangements in additive number theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Game Theory and Economic Paradigms: Imperfect Information

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Graham conjectured in 1971 that for any prime $p$, any subset $S\subseteq \mathbb{Z}_p\setminus \{0\}$ admits an ordering $s_1,s_2,\dots,s_{|S|}$ where all partial sums $s_1, s_1+s_2,\dots,s_1+s_2+\dots+s_{|S|}$ are distinct. We prove this conjecture for all subsets $S\subseteq \mathbb{Z}_p\setminus \{0\}$ with $|S|\le p^{1-α}$ and $|S|$ sufficiently large with respect to $α$, for any $α\in (0,1)$. Combined with earlier results, this gives a complete resolution of Graham's rearrangement conjecture for all sufficiently large primes $p$.

Problem

• Graham's conjecture
• rearrangement
• partial sums
• distinct sums
• finite fields

Innovation

• Graham's rearrangement conjecture
• distinct partial sums
• subset ordering
• finite fields
• combinatorial number theory