erdos_541

Let $a_1, \dots, a_p$ be (not necessarily distinct) residues modulo a prime $p$, such that there exists some $r$ so that if $S \subseteq [p]$ is non-empty and $$\sum_{i \in S} a_i \equiv 0 \pmod{p}$$ then $|S| = r$. Must there be at most two distinct residues amongst the $a_i$?

theorem erdos_541 : answer(True) ↔ (∀ p, Fact p.Prime → ∀ (a : Fin p → ZMod p),
    (∃ r, ∀ (S : Finset (Fin p)), S ≠ ∅ → ∑ i ∈ S, a i = 0 → S.card = r) →
      (Set.range a).ncard ≤ 2)

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