FALSIFIABLE
Open, but could be disproved with a finite counterexample.
Let $p$ be a prime. Given any finite set $A\subseteq \mathbb{F}_p\backslash \{0\}$, is there always a rearrangement $A=\{a_1,\ldots,a_t\}$ such that all partial sums $\sum_{1\leq k\leq m}a_{k}$ are distinct, for all $1\leq m\leq t$?
A problem of Graham, who proved it when $t=p-1$. A similar conjecture was made for arbitrary abelian groups by Alspach. Such an ordering is often called a valid ordering.
This has been proved for $t\leq 12$ (see Costa and Pellegrini
[CoPe20] and the references therein) and for $p-3\leq t\leq p-1$ (see Hicks, Ollis, and Schmitt
[HOS19] and the references therein). Kravitz
[Kr24] has proved this for\[t \leq \frac{\log p}{\log\log p}.\](This was independently earlier observed by Will Sawin in a
MathOverflow post.)
Bedert and Kravitz
[BeKr24] have now proved this conjecture for\[t \leq e^{(\log p)^{1/4}}.\]
View the LaTeX source
Additional thanks to: Zachary Chase and Noah Kravitz
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #475, https://www.erdosproblems.com/475, accessed 2026-01-16
