On Graham's rearrangement conjecture
2026-02-17 • Discrete Mathematics
Discrete Mathematics
AI summaryⓘ
The authors study a problem proposed by Graham in 1971 about arranging numbers in a special order so that all running totals are different. They prove the conjecture is true for certain smaller subsets of numbers modulo a prime number when the subsets are large enough but not too big. Their result, combined with previous work, solves the conjecture completely for all large prime numbers. This advances our understanding of how numbers can be rearranged with unique partial sums modulo primes.
Graham conjectureprime numbersmodular arithmeticsubset orderingpartial sumscombinatoricsnumber theoryfinite fieldspermutations
Authors
Huy Tuan Pham, Lisa Sauermann
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 $p$.