erdős #11
Is every large odd integer the sum of a squarefree number and a power of 2?
Worked, still open.
number theory · open · formalized (Lean) · 0 attempts
machinery: Wieferich-primes,squarefree-plus-power-of-2,additive-basis,prime-distribution,covering-congruences,sieve/Brun-Titchmarsh
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vela registry pull vfr_37aec80d874a0239vela reproduce examples/erdos-problemsevidence
unverified AI candidates (2)
gpt-erdos · GPT-5.2 Pro + Deep Research · unverified
This is **not known** in full generality. It’s an **Erdős conjecture** (often listed as “Erdős Problem #11”): for every **odd** integer (n>1), there should exist an integer (k\ge 0) such that [ n = s + 2^k ] with $$s$$ **squarefree**.
candidate solution ↗llm-hunter · codex 5.2 extra high, gpt 5.2, gpt pro 5.2 · unverified
5 LLM attack(s) recorded (codex 5.2 extra high, gpt 5.2, gpt pro 5.2); unverified.
candidate solution ↗formal
AMS 11 · open (literature)
theorem erdos_11 (n : ℕ) (hn : Odd n) (hn' : 1 < n) :
∃ k l : ℕ, Squarefree k ∧ n = k + 2 ^ lformal-conjectures/11.lean ↗oeis
A001220 — Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1.1093,3511A377587 — a(n) is the smallest odd integer m with m-2^k not squarefree for all 1<=k<=n.11,29,533,849,434977,10329791,28819433,129747557,6915752957,2569472629649,23373845739407,60690478781437
links
Wieferich primes · reference
status
open