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Constraint Satisfaction and Optimization: Satisfiability Modulo TheoriesReasoning under Uncertainty: Other Foundations of Reasoning under UncertaintyGame Theory and Economic Paradigms: Imperfect Information
On Graham's rearrangement conjecture
🤖 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.
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Application Category
📝 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
Research questions and friction points this paper is trying to address.
Graham's conjecture
rearrangement
partial sums
distinct sums
finite fields
Innovation
Methods, ideas, or system contributions that make the work stand out.
Graham's rearrangement conjecture
distinct partial sums
subset ordering
finite fields
combinatorial number theory
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