541.lean`

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Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

3 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 541

References:

erdosproblems.com/541
[ErSz76] Erdős, E. and Szemerédi, E., On a problem of Graham. Publ. Math. Debrecen (1976), 123--127.
[GHW10] Gao, Weidong and Hamidoune, Yahya Ould and Wang, Guoqing, Distinct length modular zero-sum subsequences: a proof of Graham's conjecture. J. Number Theory (2010), 1425--1431.

Imports (1)

FormalConjecturesUtil

theorem Erdos541.erdos_541 Documented declaration: erdos_541 sorry line 48 source
Source documentation
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 $a_i$?
This was formalized in Lean by Alexeev using Aristotle and ChatGPT.

depends on sorry what this means
Statement uses: Iff, True, Nat, Fact, Nat.Prime, Fin, ZMod, Exists, Finset, Ne, EmptyCollection.emptyCollection, Eq, Finset.sum, OfNat.ofNat, Finset.card, LE.le, Set.ncard, Set.range
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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) := by
…
```
theorem Erdos541.erdos_541.variants.general_moduli Gao, Hamidoune, and Wang [GHW10] solved this for all moduli p (not necessarily prime) sorry line 55 source
Source documentation
Gao, Hamidoune, and Wang [GHW10] solved this for all moduli p (not necessarily prime).

depends on sorry what this means
Statement uses: Nat, Fin, ZMod, Exists, Finset, Ne, EmptyCollection.emptyCollection, Eq, Finset.sum, OfNat.ofNat, Finset.card, LE.le, Set.ncard, Set.range
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_541.variants.general_moduli (p : ℕ) (a : Fin p → ZMod p)
    (ha₀ : ∃ r, ∀ (S : Finset (Fin p)), S ≠ ∅ → ∑ i ∈ S, a i = 0 → S.card = r) :
      (Set.range a).ncard ≤ 2 := by
…
```
theorem Erdos541.erdos_541.variants.large_primes This was proved by Erdős and Szemerédi [ErSz76] for p sufficiently large sorry line 62 source
Source documentation
This was proved by Erdős and Szemerédi [ErSz76] for p sufficiently large.

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, Nat.Prime, Fin, ZMod, Exists, Finset, Ne, EmptyCollection.emptyCollection, Eq, Finset.sum, OfNat.ofNat, Finset.card, LE.le, Set.ncard, Set.range, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_541.variants.large_primes : ∀ᶠ p in atTop, 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 := by
…
```