Is there a set $A \subset N$ such that $∣ A \cap {1, \dots, N}∣ = o ((lo g N)^{2})$ and such that every large integer can be written as $p + a$ for some prime $p$ and $a \in A$ ? Can the bound $O (lo g N)$ be achieved? Must such an $A$ satisfy $lim inf \frac{∣ A \cap { 1 , \dots , N }∣}{lo g N} > 1 ?$

unreviewedOpen. Worked here; no verified result yet.

number theory · open · formalized (Lean) · 0 attempts

machinery: additive-complement,additive-basis,prime-distribution,sieve/Brun-Titchmarsh,probabilistic-covering,Hardy-Littlewood,additive-combinatorics

Check This Result Yourself

use this data

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

Evidence

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let (P) denote the primes, and write the counting function [ A(N):=\lvert A\cap{1,\dots,N}\rvert . ] Your condition is that (P+A) contains all sufficiently large integers (i.e. all but finitely many).

candidate solution ↗

llm-hunter · gpt 5.2, gpt pro 5.2 · unverified

3 LLM attack(s) recorded (gpt 5.2, gpt pro 5.2); unverified.

candidate solution ↗

Formal proof

AMS 11 · solved (literature)

theorem erdos_32.variants.log_squared : ∃ A : Set ℕ,
    IsAdditiveComplementToPrimes A ∧
    (fun N => (((Finset.Icc 1 N).filter (· ∈ A)).card : ℝ)) =O[atTop]
      fun N => (Real.log N) ^ 2

formal-conjectures/32.lean ↗

Check it yourself

One command re-derives this record's receipts on your machine.

vela reproduce examples/erdos-problems

Search Vela