682.lean`

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

4 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 682

References:

erdosproblems.com/682
[Er79d] Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.
[GaTa25] A. Gafni and T. Tao, Rough numbers between consecutive primes. arXiv:2508.06463 (2025).

Imports (1)

FormalConjecturesUtil

abbrev Erdos682.p The nth prime, indexed from zero: p 0 = 2, p 1 = 3, line 37 source
Source documentation
The nth prime, indexed from zero: p 0 = 2, p 1 = 3, ...

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.nth, Nat.Prime
```
noncomputable abbrev p (n : ℕ) : ℕ := Nat.nth Nat.Prime n
…
```
theorem Erdos682.erdos_682 Is it true that for almost all $n$ there exists some $m\in (p_n,p_{n+1})$ such that $$p(m) \geq p_{n+1}-p_n,$$ where $p(m)$ denotes the least prime factor of $m$? sorry line 49 source
Source documentation
Is it true that for almost all $n$ there exists some $m\in (p_n,p_{n+1})$ such that $$p(m) \geq p_{n+1}-p_n,$$ where $p(m)$ denotes the least prime factor of $m$?
This was solved in the affirmative by Gafni and Tao [GaTa25].
This was formalized in Lean by Codex and GPT-5.6 Sol.

depends on sorry what this means
Statement uses: Iff, True, Set.HasDensity, Nat, Set.ofPred, Exists, And, Membership.mem, Set, Set.Ioo, Erdos682.p, HAdd.hAdd, OfNat.ofNat, LE.le, HSub.hSub, Nat.minFac, Real, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_682 : answer(True) ↔
    {n | ∃ m ∈ Set.Ioo (p n) (p (n + 1)), p (n + 1) - p n ≤ m.minFac}.HasDensity 1 := by
…
```
Mentions: Erdos682.p
theorem Erdos682.erdos_682.variants.exceptional_count Gafni and Tao [GaTa25, Theorem 1.1] proved that the number of exceptional gaps $(p_n,p_{n+1})$ with $p_n\in [X,2X]$ is $$\ll \frac{X}{(\log X)^2}.$$ sorry line 59 source
Source documentation
Gafni and Tao [GaTa25, Theorem 1.1] proved that the number of exceptional gaps $(p_n,p_{n+1})$ with $p_n\in [X,2X]$ is $$\ll \frac{X}{(\log X)^2}.$$

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Set.ncard, Set.ofPred, And, Membership.mem, Set, Set.Icc, HMul.hMul, OfNat.ofNat, Erdos682.p, Set.Ioo, HAdd.hAdd, LT.lt, Nat.minFac, HSub.hSub, HDiv.hDiv, HPow.hPow and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_682.variants.exceptional_count :
    (fun X : ℕ ↦ ({n | p n ∈ Set.Icc X (2 * X) ∧
        ∀ m ∈ Set.Ioo (p n) (p (n + 1)), m.minFac < p (n + 1) - p n}.ncard : ℝ)) =O[atTop]
      fun X : ℕ ↦ (X : ℝ) / Real.log X ^ 2 := by
…
```
Mentions: Erdos682.p
theorem Erdos682.erdos_682.variants.dickson_counterexample Erdős first thought this should be true for all large sorry line 76 source
Source documentation
Erdős first thought this should be true for all large $n$, but found a (conditional) counterexample: Dickson's conjecture says there are infinitely many $d$ such that $$2183+30030d\textrm{ and }2201+30030d$$ are both prime, and then they must necessarily be consecutive primes. These give a counterexample since $30030=2\cdot 3 \cdot 5\cdot 7\cdot 11\cdot 13$ and every integer in $[2184,2200]$ is divisible by at least one of these primes.
This was formalized in Lean by Codex and GPT-5.6 Sol.

depends on sorry what this means
Statement uses: Nat, Nat.Prime, HAdd.hAdd, OfNat.ofNat, HMul.hMul, Exists, And, Eq, Erdos682.p, Membership.mem, Set, Set.Ioo, LT.lt, Nat.minFac, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_682.variants.dickson_counterexample (d : ℕ) (h₁ : (2183 + 30030 * d).Prime)
    (h₂ : (2201 + 30030 * d).Prime) : ∃ n, p n = 2183 + 30030 * d ∧ p (n + 1) = 2201 + 30030 * d ∧
      ∀ m ∈ Set.Ioo (p n) (p (n + 1)), m.minFac < p (n + 1) - p n := by
…
```
Mentions: Erdos682.p