699.lean`

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L4DEL: Lean4 Domain Explanation Language

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

3 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 699

Reference: erdosproblems.com/699

Imports (1)

FormalConjecturesUtil

theorem Erdos699.sylvester_schur Sylvester and Schur: for $1 \le i \le n/2$ there is a prime $p > i$ dividing n.choose i sorry line 32 source
Source documentation
Sylvester and Schur: for $1 \le i \le n/2$ there is a prime $p > i$ dividing n.choose i.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, HDiv.hDiv, Exists, And, Nat.Prime, LT.lt, Dvd.dvd, Nat.choose
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem sylvester_schur (n i : ℕ) (hi : 1 ≤ i) (hi_half : i ≤ n / 2) :
    ∃ p : ℕ, p.Prime ∧ i < p ∧ p ∣ Nat.choose n i := by
…
```
theorem Erdos699.erdos_699 Erdős Problem 699 sorry line 41 source
Source documentation
Erdős Problem 699. Is it true that for every $1 \le i < j \le n / 2$ there exists a prime $p \ge i$ with $p \mid \gcd\big(\binom{n}{i}, \binom{n}{j}\big)$?

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, LT.lt, HDiv.hDiv, Exists, And, Nat.Prime, Dvd.dvd, Nat.gcd, Nat.choose
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_699 : answer(sorry) ↔
    ∀ n i j : ℕ,
      1 ≤ i →
      i < j →
      j ≤ n / 2 →
      ∃ p : ℕ, p.Prime ∧ i ≤ p ∧ p ∣ Nat.gcd (Nat.choose n i) (Nat.choose n j) := by
…
```
theorem Erdos699.erdos_szekeres_strengthening Erdős and Szekeres conjectured that, apart from a finite exceptional set of triples (n, i, j), one can always take p > i in the prime divisor statement sorry line 52 source
Source documentation
Erdős and Szekeres conjectured that, apart from a finite exceptional set of triples (n, i, j), one can always take p > i in the prime divisor statement.

depends on sorry what this means
Statement uses: Iff, True, Exists, Finset, Prod, Nat, LE.le, OfNat.ofNat, LT.lt, HDiv.hDiv, Not, Membership.mem, Prod.mk, And, Nat.Prime, Dvd.dvd, Nat.gcd, Nat.choose
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_szekeres_strengthening : answer(sorry) ↔
    ∃ E : Finset (ℕ × ℕ × ℕ), ∀ n i j : ℕ,
      1 ≤ i →
      i < j →
      j ≤ n / 2 →
      (n, i, j) ∉ E →
      ∃ p : ℕ, p.Prime ∧ i < p ∧ p ∣ Nat.gcd (Nat.choose n i) (Nat.choose n j) := by
…
```