453.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

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 453

References:

erdosproblems.com/453
[Er70b] Erdős, P., Some applications of graph theory to number theory. Proc. Second Chapel Hill Conf. on Combinatorial Mathematics and its Applications (Univ. North Carolina, Chapel Hill, N.C., 1970) (1970), 136-145.
[Er74b] Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.
[Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
[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.
[Po79] Pomerance, Carl, The prime number graph. Math. Comp. (1979), 399-408.

Imports (1)

FormalConjecturesUtil

def Erdos453.EventuallyHasPrimeWitness The eventual prime inequality asked in Erdős Problem 453, using Mathlib's zero-based primes line 44 source
Source documentation
The eventual prime inequality asked in Erdős Problem 453, using Mathlib's zero-based primes.

standard axioms: Classical.choice, Quot.sound, propext what this means
Proof uses: Exists, Nat, LE.le, And, LT.lt, HPow.hPow, Nat.nth, Nat.Prime, OfNat.ofNat, HMul.hMul, HAdd.hAdd, HSub.hSub
13 instances, coercions and auxiliary constants left out.
```
def EventuallyHasPrimeWitness : Prop :=
…
```
theorem Erdos453.erdos_453 Is it true that, for all sufficiently large $n$, there exists some $i<n$ such that $$ p_n^2 < p_{n+i}p_{n-i}, $$ where $p_k$ is the $k$th prime? sorry line 60 source
Source documentation
Is it true that, for all sufficiently large $n$, there exists some $i<n$ such that $$ p_n^2 < p_{n+i}p_{n-i}, $$ where $p_k$ is the $k$th prime?
Pomerance proved that the answer is no.

depends on sorry what this means
Statement uses: Iff, False, Erdos453.EventuallyHasPrimeWitness
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_453 : answer(False) ↔ EventuallyHasPrimeWitness := by
…
```
Mentions: Erdos453.EventuallyHasPrimeWitness