413.lean`

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7 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 413

References:

erdosproblems.com/413
A5236
[La26] Lau, C. F., On the Number of Prime Factors of Consecutive Integers. arXiv:2604.15042 (2026).

Erdős called a natural number n a barrier for ω, the number of distinct prime divisors, if m + ω(m) ≤ n for all m < n. He believed there should be infinitely many such barriers, and even posed a relaxed variant asking whether there is some ε > 0 for which infinitely many n satisfy m + ε · ω(m) ≤ n for every m < n.

Imports (1)

FormalConjecturesUtil

def Erdos413.IsBarrier IsBarrier f n means n is a barrier for the real-valued function f, i.e. (m : ℝ) + f m ≤ (n : ℝ) for all m < n line 44 source
Source documentation
IsBarrier f n means n is a barrier for the real-valued function f, i.e. (m : ℝ) + f m ≤ (n : ℝ) for all m < n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: LT.lt, LE.le, HAdd.hAdd
6 instances, coercions and auxiliary constants left out.
```
def IsBarrier (f : ℕ → ℝ) (n : ℕ) : Prop :=
…
```
theorem Erdos413.erdos_413.parts.i Are there infinitely many barriers for ω? sorry line 49 source
Source documentation
Are there infinitely many barriers for ω?

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, Erdos413.IsBarrier, Real, ArithmeticFunction, ArithmeticFunction.cardDistinctFactors
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_413.parts.i :
    answer(sorry) ↔ { n | IsBarrier (fun m => ω m) n }.Infinite := by
…
```
Mentions: Erdos413.IsBarrier
def Erdos413.expProd expProd n is ∏ kᵢ when n = ∏ pᵢ ^ kᵢ, i.e. the product of the prime exponents of n line 54 source
Source documentation
expProd n is ∏ kᵢ when n = ∏ pᵢ ^ kᵢ, i.e. the product of the prime exponents of n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finsupp.prod, Nat.factorization
3 instances, coercions and auxiliary constants left out.
```
def expProd (n : ℕ) : ℕ :=
…
```
theorem Erdos413.erdos_413.variants.hasPosDensity_barrier_expProd Erdős proved that the barrier set for expProd is infinite and even has positive density sorry line 69 source
Source documentation
Erdős proved that the barrier set for expProd is infinite and even has positive density.
HasPosDensity is the right reading rather than positive lower density. In [Er79d] this is Theorem 1, "the density of integers satisfying (2) is positive", where d₀(n) = ∏ αᵢ is expProd. The averaging argument there bounds the density below, but Erdős states the existence separately on the last page: "With a little more trouble, I can prove that the density of integers n for which n is a barrier for d₀(n) exists." He goes further, that if αᵢ is the density of n with max_{m<n} (m + d₀(m)) = n + i, then every αᵢ exists and they sum to 1.
[Er79d] Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.

depends on sorry what this means
Statement uses: Set.HasPosDensity, Nat, Set.ofPred, Erdos413.IsBarrier, Real, Erdos413.expProd, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_413.variants.hasPosDensity_barrier_expProd :
    { n | IsBarrier (fun m => expProd m) n }.HasPosDensity := by
…
```
Mentions: Erdos413.IsBarrier, Erdos413.expProd
theorem Erdos413.erdos_413.variants.bigOmega Erdős believed there should be infinitely many barriers for Ω, the total prime multiplicity sorry line 75 source
Source documentation
Erdős believed there should be infinitely many barriers for Ω, the total prime multiplicity.

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, Erdos413.IsBarrier, Real, ArithmeticFunction, ArithmeticFunction.cardFactors
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_413.variants.bigOmega :
    answer(sorry) ↔ { n | IsBarrier (fun m => Ω m) n }.Infinite := by
…
```
Mentions: Erdos413.IsBarrier
theorem Erdos413.erdos_413.variants.bigOmega_largest_barrier_lt_100k Selfridge computed that the largest Ω-barrier below 10^5 is 99840 sorry line 81 source
Source documentation
Selfridge computed that the largest Ω-barrier below 10^5 is 99840.

depends on sorry what this means
Statement uses: IsGreatest, Nat, Set.ofPred, And, LT.lt, HPow.hPow, OfNat.ofNat, Erdos413.IsBarrier, Real, ArithmeticFunction, ArithmeticFunction.cardFactors
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_413.variants.bigOmega_largest_barrier_lt_100k :
    IsGreatest {n : ℕ | n < 10 ^ 5 ∧ IsBarrier (fun m => Ω m) n} 99840 := by
…
```
Mentions: Erdos413.IsBarrier
theorem Erdos413.erdos_413.parts.ii Does there exist some $\epsilon > 0$ such that there are infinitely many $\epsilon$-barriers for $\omega$? sorry line 88 source
Source documentation
Does there exist some $\epsilon > 0$ such that there are infinitely many $\epsilon$-barriers for $\omega$? Lau [La26, Theorem 1.3] gives a positive answer.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Set.Infinite, Nat, Set.ofPred, Erdos413.IsBarrier, HMul.hMul, ArithmeticFunction, ArithmeticFunction.cardDistinctFactors
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_413.parts.ii :
    answer(True) ↔
        (∃ ε > (0 : ℝ), { n | IsBarrier (fun n => ε * ω n) n }.Infinite) := by
…
```
Mentions: Erdos413.IsBarrier