Erdős Problem 41

formal-conjectures · erdos-problems · ams-11

Let $A \subset \mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct
for $a,b,c \in A$ (aside from the trivial coincidences). Is it true that
$$\liminf_{N \to \infty} \frac{\lvert A \cap \{1,\ldots,N\}\rvert}{N^{1/3}}=0?$$

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://www.erdosproblems.com/41
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/ErdosProblems/41.lean
Local target: Corpus.Erdos41.erdos_41
Source SHA-256: 3c3f748438f05e1b9acf1b529dabe5bd976c0658cf305a5560dcedbcee770029
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record