1148.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

6 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1148

References:

erdosproblems.com/1148
[Ch26] P. Chojecki, Bounded Representations by $x^2 + y^2 - z^2$ (2026)
[Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999).

Imports (1)

FormalConjecturesUtil

def Erdos1148.Erdos1148Prop A natural number $n$ which can be written as $n$ if $n = x^2 + y^2 - z^2$ with $\max(x^2, y^2, z^2) \leq n$ line 40 source
Source documentation
A natural number $n$ which can be written as $n$ if $n = x^2 + y^2 - z^2$ with $\max(x^2, y^2, z^2) \leq n$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: Exists, And, Eq, HSub.hSub, HAdd.hAdd, HPow.hPow, OfNat.ofNat, LE.le
10 instances, coercions and auxiliary constants left out.
```
def Erdos1148Prop (n : ℕ) : Prop :=
…
```
theorem Erdos1148.erdos_1148 sorry line 55 source
depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, Erdos1148.Erdos1148Prop, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
instance Erdos1148.(instance at line 61) line 61 source
Source documentation
The largest integer known which cannot be written this way is $6563$.
```
private instance (n : ℕ) : Decidable (Erdos1148Prop n) :=
…
```
Mentions: Erdos1148.Erdos1148Prop
theorem Erdos1148.erdos_1148.variants.lower_bound The integer $6563$ cannot be written as $x^2 + y^2 - z^2$ with $\max(x^2, y^2, z^2) \leq 6563$ line 78 source
Source documentation
The integer $6563$ cannot be written as $x^2 + y^2 - z^2$ with $\max(x^2, y^2, z^2) \leq 6563$.

extra axioms: _private.0.Erdos1148.erdos_1148.variants.lower_bound._native.decide.ax_1 what this means
Statement uses: Not, Erdos1148.Erdos1148Prop, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1148.variants.lower_bound : ¬ Erdos1148Prop 6563 := by
…
```
Mentions: Erdos1148.Erdos1148Prop
def Erdos1148.erdos_1148_weaker_prop The weaker property: $n = x^2 + y^2 - z^2$ such that $\max(x^2, y^2, z^2) \leq n + 2\sqrt{n}$ line 84 source
Source documentation
The weaker property: $n = x^2 + y^2 - z^2$ such that $\max(x^2, y^2, z^2) \leq n + 2\sqrt{n}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Exists, And, Eq, HSub.hSub, HAdd.hAdd, HPow.hPow, OfNat.ofNat, LE.le, Real, HMul.hMul, Real.sqrt
18 instances, coercions and auxiliary constants left out.
```
def erdos_1148_weaker_prop (n : ℕ) : Prop :=
…
```
theorem Erdos1148.erdos_1148.variants.weaker [Va99] reports this is 'obvious' if we replace $\leq n$ with $\leq n+2\sqrt{n}$ sorry line 94 source
Source documentation
[Va99] reports this is 'obvious' if we replace $\leq n$ with $\leq n+2\sqrt{n}$.

depends on sorry what this means
Statement uses: Nat, Erdos1148.erdos_1148_weaker_prop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1148.variants.weaker : ∀ n, erdos_1148_weaker_prop n := by
…
```
Mentions: Erdos1148.erdos_1148_weaker_prop