851.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

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

4 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 851

References:

erdosproblems.com/851
[Pr26] D. Price and GPT-5.2 Pro, Erdős problem 851 (2026)

Imports (1)

FormalConjecturesUtil

def Erdos851.TwoPowAddSet TwoPowAddSet r is the set of integers of the form 2^k+n, where k ≥ 0 and n has at most r prime divisors line 36 source
Source documentation
TwoPowAddSet r is the set of integers of the form 2^k+n, where k ≥ 0 and n has at most r prime divisors.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set
Proof uses: Set.ofPred, Exists, LE.le, Finset.card, Nat.primeFactors, Eq, HAdd.hAdd, HPow.hPow, OfNat.ofNat
8 instances, coercions and auxiliary constants left out.
```
def TwoPowAddSet (r : ℕ) := {(2 ^ k + n) | (k : ℕ) (n : ℕ) (_ : n.primeFactors.card ≤ r)}
```
def Erdos851.twoPowAddPrimeSet The set of integers of the form 2^k+p, where k ≥ 0 and p is prime line 39 source
Source documentation
The set of integers of the form 2^k+p, where k ≥ 0 and p is prime.

standard axioms: propext what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, Exists, Nat.Prime, Eq, HAdd.hAdd, HPow.hPow, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.
```
def twoPowAddPrimeSet : Set ℕ := {(2 ^ k + p) | (k : ℕ) (p : ℕ) (_ : p.Prime)}
…
```
theorem Erdos851.erdos_851.variants.romanoff The set of integers of the form 2^k+p (where p is prime) has positive lower density sorry line 45 source
Source documentation
The set of integers of the form 2^k+p (where p is prime) has positive lower density.

depends on sorry what this means
Statement uses: LT.lt, Real, OfNat.ofNat, Set.lowerDensity, Nat, Erdos851.twoPowAddPrimeSet, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_851.variants.romanoff : 0 < Set.lowerDensity twoPowAddPrimeSet := by
…
```
Mentions: Set.lowerDensity, Erdos851.twoPowAddPrimeSet
theorem Erdos851.erdos_851 sorry line 56 source
depends on sorry what this means
Statement uses: Real, Membership.mem, Set, Set.Ioo, OfNat.ofNat, Exists, Nat, LE.le, HSub.hSub, Set.lowerDensity, Erdos851.TwoPowAddSet, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.