457.lean`

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

4 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 457

Reference: erdosproblems.com/457

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

theorem Erdos457.erdos_457 Is there some sorry line 40 source
Source documentation
Is there some $\epsilon > 0$ such that there are infinitely many $n$ where all primes $p \le (2 + \epsilon) \log n$ divide $$ \prod_{1 \le i \le \log n} (n + i)? $$
This was formalized in Lean by Baretto and van Doorn using Aristotle.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Set.Infinite, Nat, Set.ofPred, Erdos457.erdos_457.match_1, LE.le, HMul.hMul, HAdd.hAdd, Real.log, Nat.Prime, Dvd.dvd, Finset.prod, Finset.Icc, Nat.floor
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
50 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_457 : answer(True) ↔ ∃ ε > (0 : ℝ),
    { (n : ℕ) | ∀ (p : ℕ), p ≤ (2 + ε) * Real.log n → p.Prime →
      p ∣ ∏ i ∈ Finset.Icc 1 ⌊Real.log n⌋₊, (n + i) }.Infinite := by
…
```
abbrev Erdos457.q Let $q(n, k)$ denote the least prime which does not divide $\prod_{1 \le i \le k}(n + i)$ line 47 source
Source documentation
Let $q(n, k)$ denote the least prime which does not divide $\prod_{1 \le i \le k}(n + i)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Nat.find, And, Nat.Prime, Not, Dvd.dvd, Finset.prod, Finset.Icc, OfNat.ofNat, Nat.floor, HAdd.hAdd
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
noncomputable abbrev q (n : ℕ) (k : ℝ) : ℕ :=
…
```
theorem Erdos457.erdos_457.variants.qnk More generally, let $q(n, k)$ denote the least prime which does not divide $\prod_{1 \le i \le k}(n + i)$ line 62 source
Source documentation
More generally, let $q(n, k)$ denote the least prime which does not divide $\prod_{1 \le i \le k}(n + i)$. This problem asks whether $q(n, \log n) \ge (2 + \epsilon) \log n$ infinitely often.
This follows from erdos_457: if every prime $p \le (2 + \epsilon) \log n$ divides $\prod_{1 \le i \le \log n} (n + i)$, then $q(n, \log n)$, the least prime not dividing this product, must exceed $(2 + \epsilon) \log n$.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Set.Infinite, Nat, Set.ofPred, Erdos457.erdos_457.match_1, LE.le, HMul.hMul, HAdd.hAdd, Real.log, Erdos457.q
Proof uses: Iff.intro, Nat.Prime, Dvd.dvd, Finset.prod, Finset.Icc, Nat.floor, Iff.mp, Erdos457.erdos_457, trivial, Exists.intro, And.intro, Set.Infinite.mono, Membership.mem, Set, Decidable.byContradiction, Not, False, Nat.find_spec, Nat.exists_prime_not_dvd, Iff.mpr and 20 more
91 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_457.variants.qnk : answer(True) ↔ ∃ ε > (0 : ℝ),
    { (n : ℕ) | (2 + ε) * Real.log n ≤ q n (Real.log n) }.Infinite := by
…
```
Mentions: Erdos457.q
theorem Erdos457.erdos_457.variants.one_sub Taking sorry line 86 source
Source documentation
Taking $n$ to be the product of primes between $\log n$ and $(2 + o(1)) \log n$ gives an example where $$ q(n, \log n) \ge (2 + o(1)) \log n. $$ Can one prove that $q(n, \log n) < (1 - \epsilon) (\log n)^2$ for all large $n$ and some $\epsilon > 0$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, LT.lt, Erdos457.q, Real.log, HMul.hMul, HSub.hSub, HPow.hPow, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_457.variants.one_sub : answer(sorry) ↔ ∃ ε > (0 : ℝ),
    ∀ᶠ n in Filter.atTop, q n (Real.log n) < (1 - ε) * Real.log n ^ 2 := by
…
```
Mentions: Erdos457.q