730.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

4 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 730

References:

erdosproblems.com/730
A129515
PALOMAR-2026-08-22-000001: a Lean 4 proof that there are infinitely many such pairs, checked by Comparator against the statement S.Infinite below and registered with the Palomar registry. The proof follows an argument posted on the erdosproblems.com forum (Liam Price, 24 June 2026), which shows that there are infinitely many consecutive pairs $(n, n+1)$.

Imports (1)

FormalConjecturesUtil

abbrev Erdos730.S line 38 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Prod, Nat
Proof uses: Set.ofPred, Erdos730.S.match_1, And, LT.lt, Eq, Finset, Nat.primeFactors, Nat.centralBinom
1 instances, coercions and auxiliary constants left out.
theorem Erdos730.erdos_730 sorry line 51 source
depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Prod, Nat, Erdos730.S
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 Erdos730.erdos_730.variants.explicit_pairs For example, $(87,88)$ and $(607,608)$ are such pairs line 58 source
Source documentation
For example, $(87,88)$ and $(607,608)$ are such pairs.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LE.le, Set, Prod, Nat, Insert.insert, Prod.mk, OfNat.ofNat, Singleton.singleton, Erdos730.S
Proof uses: Membership.mem, Eq, And.intro, LT.lt, Finset, Nat.primeFactors, Nat.centralBinom, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, HDiv.hDiv, Nat.descFactorial, HMul.hMul, Nat.factorial, congr, congrArg and 2 more
18 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_730.variants.explicit_pairs :
    {(87, 88), (607, 608)} ⊆ S := by
…
```
Mentions: Erdos730.S
theorem Erdos730.erdos_730.variants.delta_ne_one There are examples where $(n, m) ∈ S$ with $m ≠ n + 1$ line 71 source
Source documentation
There are examples where $(n, m) ∈ S$ with $m ≠ n + 1$.
(Found by AlphaProof, although it was implicit already in [A129515])

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Exists, Nat, And, Membership.mem, Prod, Set, Erdos730.S, Prod.mk, Ne, HAdd.hAdd, OfNat.ofNat
Proof uses: id, Exists.intro, LT.lt, Eq, Finset, Nat.primeFactors, Nat.centralBinom, Not, Eq.mpr, Nat.Prime, Iff, Dvd.dvd, Nat.choose, Eq.trans, True, congr, congrArg, eq_true, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat and 31 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_730.variants.delta_ne_one : ∃ (n m : ℕ), (n, m) ∈ S ∧ m ≠ n + 1 := by
…
```
Mentions: Erdos730.S