FormalConjectures.ErdosProblems.«476»

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import FormalConjecturesUtil

Erdős Problem 476

References:

erdosproblems.com/476

[Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.

[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.

[dSHa94] Dias da Silva, J. A. and Hamidoune, Y. O., Cyclic spaces for Grassmann derivatives and additive theory. Bull. London Math. Soc. (1994), 140-146.


namespace Erdos476

Let $A\subseteq \mathbb{F}_p$. Let $$ A\hat{+}A = { a+b : a\neq b \in A}. $$ Is it true that $$ \lvert A\hat{+}A\rvert \geq \min(2\lvert A\rvert-3,p)? $$

This is the Erdős-Heilbronn inequality, proved by Dias da Silva and Hamidoune.

@[category research solved, AMS 5 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos476.lean"]
theorem declaration uses 'sorry'erdos_476 : answer(True) ↔
    ∀ p : ℕ, Fact p.Prime → ∀ A : Finset (ZMod p),
      A.restrictedSumset.card ≥ min (2 * A.card - 3) p := by⊢ True ↔ ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (A : Finset (ZMod p)), A.restrictedSumset.card ≥ min (2 * A.card - 3) p
  sorryAll goals completed! 🐙


end Erdos476