365.lean`

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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

3 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 365

References:

erdosproblems.com/365
[Go70] Golomb, S. W., Powerful numbers. Amer. Math. Monthly (1970), 848-855.
[Gu04] Guy, R. K., Unsolved problems in number theory (2004), Problem B16.
[Wa76] Walker, D. T., Consecutive integer pairs of powerful numbers and related Diophantine equations. Fibonacci Quart. (1976), 111-116.

Imports (1)

FormalConjecturesUtil

theorem Erdos365.golomb_pair Golomb's consecutive powerful pair $12167=23^3$ and $12168=2^3 3^2 13^2$ line 38 source
Source documentation
Golomb's consecutive powerful pair $12167=23^3$ and $12168=2^3 3^2 13^2$. Neither number is a square [Go70].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: And, Nat.Powerful, OfNat.ofNat, Nat, Not, IsSquare
Proof uses: Membership.mem, Finset, Nat.primeFactors, Nat.Prime, Dvd.dvd, Ne, HPow.hPow, Iff.mp, Nat.mem_primeFactors, Eq.mpr, id, Eq, congrArg, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_pow, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Meta.NormNum.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.bit1, Nat.prime_eq_prime_of_dvd_pow and 86 more
115 instances, coercions and auxiliary constants left out.

Tactic trace: 46 steps
Open to load the checked before-and-after goal states.
```
theorem golomb_pair :
    Nat.Powerful 12167 ∧ Nat.Powerful 12168 ∧
      ¬ IsSquare (12167 : ℕ) ∧ ¬ IsSquare (12168 : ℕ) := by
…
```
Mentions: Nat.Powerful
theorem Erdos365.erdos_365.parts.i Do all pairs of consecutive powerful numbers $n$ and $n+1$ come from solutions to Pell equations? line 80 source
Source documentation
Do all pairs of consecutive powerful numbers $n$ and $n+1$ come from solutions to Pell equations? In other words, must either $n$ or $n+1$ be a square?
The answer to the first question is no: Golomb [Go70] observed that both $12167=23^3$ and $12168=2^3 3^2 13^2$ are powerful.
We require $n>0$ since Nat.Powerful also includes $0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Iff, False, Nat, LT.lt, OfNat.ofNat, Nat.Powerful, HAdd.hAdd, Or, IsSquare
Proof uses: Iff.intro, False.elim, And, Not, Erdos365.golomb_pair, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false
20 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_365.parts.i : answer(False) ↔
    ∀ n : ℕ, 0 < n → Nat.Powerful n → Nat.Powerful (n + 1) →
      IsSquare n ∨ IsSquare (n + 1) := by
…
```
Mentions: Nat.Powerful
theorem Erdos365.erdos_365.parts.ii Is the number of positive $n\leq x$ such that $n$ and $n+1$ are both powerful bounded by $(\log x)^{O(1)}$? sorry line 95 source
Source documentation
Is the number of positive $n\leq x$ such that $n$ and $n+1$ are both powerful bounded by $(\log x)^{O(1)}$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Nat, Filter.Eventually, LE.le, Set.ncard, Set.ofPred, Nat.Powerful, HAdd.hAdd, HMul.hMul, HPow.hPow, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_365.parts.ii : answer(sorry) ↔
    ∃ C : ℝ, 0 < C ∧ ∃ k : ℕ, ∀ᶠ x : ℕ in Filter.atTop,
      ({n : ℕ | 0 < n ∧ n ≤ x ∧ Nat.Powerful n ∧ Nat.Powerful (n + 1)}.ncard : ℝ) ≤
        C * (Real.log x) ^ k := by
…
```
Mentions: Nat.Powerful