Problem Statement
Let $A\subseteq \mathbb{F}_p$. Let\[A\hat{+}A = \{ a+b : a\neq b \in A\}.\]Is it true that\[\lvert A\hat{+}A\rvert \geq \min(2\lvert A\rvert-3,p)?\]
Categories:
Number Theory Additive Combinatorics
Progress
A question of Erdős and Heilbronn. Solved in the affirmative by da Silva and Hamidoune [dSHa94].In [Er65b] Erdős mentions the more general conjecture that the number of $x\in \mathbb{F}_p$ which are the sum of at most $r$ many distinct elements of $A$ is at least\[\min(r\lvert A\rvert-r^2+1,p).\]This is discussed in problem C15 of Guy's collection [Gu04].
Source: erdosproblems.com/476 | Last verified: January 15, 2026