698.lean`

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L4DEL: Lean4 Domain Explanation Language

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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

4 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 698

References:

erdosproblems.com/698
[ErSz78] Erdős, P. and Szekeres, G., Some number theoretic problems on binomial coefficients. Austral. Math. Soc. Gaz. (1978), 97-99.
[Be11] Bergman, George M., On common divisors of multinomial coefficients. Bull. Aust. Math. Soc. (2011), 138--157.

Imports (1)

FormalConjecturesUtil

theorem Erdos698.erdos_698 sorry line 51 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, Filter.Tendsto, Filter.atTop, LE.le, OfNat.ofNat, LT.lt, HDiv.hDiv, 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 Erdos698.erdos_698.variants.erdos_szekeres Documented declaration: erdos_szekeres sorry line 64 source
Source documentation
A problem of Erdős and Szekeres, who observed that $$\textrm{gcd}\left( \binom{n}{i},\binom{n}{j}\right) \geq \frac{\binom{n}{i}}{\binom{j}{i}} \geq 2^i$$ (in particular the greatest common divisor is always $>1$).

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, LT.lt, HDiv.hDiv, And, Real, Nat.choose, Nat.gcd, HPow.hPow
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_698.variants.erdos_szekeres (n i j : ℕ) (hi : 1 ≤ i) (hij : i < j)
    (hj : j ≤ n / 2) :
    (n.choose i : ℝ) / (j.choose i : ℝ) ≤ (Nat.gcd (n.choose i) (n.choose j) : ℝ) ∧
      (2 : ℝ) ^ i ≤ (n.choose i : ℝ) / (j.choose i : ℝ) := by
…
```
theorem Erdos698.erdos_698.variants.erdos_szekeres_sharp This inequality is sharp for $i=1$, $j=p$, and $n=2p$ sorry line 74 source
Source documentation
This inequality is sharp for $i=1$, $j=p$, and $n=2p$.

depends on sorry what this means
Statement uses: Nat, Nat.Prime, LT.lt, OfNat.ofNat, And, Eq, Real, Nat.gcd, Nat.choose, HMul.hMul, HDiv.hDiv, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_698.variants.erdos_szekeres_sharp (p : ℕ) (hp : p.Prime) (hp2 : 2 < p) :
    (Nat.gcd ((2 * p).choose 1) ((2 * p).choose p) : ℝ) =
        ((2 * p).choose 1 : ℝ) / (p.choose 1 : ℝ) ∧
      ((2 * p).choose 1 : ℝ) / (p.choose 1 : ℝ) = (2 : ℝ) ^ 1 := by
…
```
theorem Erdos698.erdos_698.variants.bergman Documented declaration: bergman sorry line 86 source
Source documentation
This was resolved by Bergman [Be11], who proved that for any $2\leq i<j\leq n/2$ $$\textrm{gcd}\left( \binom{n}{i},\binom{n}{j}\right) \gg n^{1/2}\frac{2^i}{i^{3/2}},$$ where the implied constant is absolute.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Nat, LE.le, HDiv.hDiv, HMul.hMul, Real.sqrt, HPow.hPow, Nat.gcd, Nat.choose
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_698.variants.bergman :
    ∃ c : ℝ, 0 < c ∧ ∀ n i j : ℕ, 2 ≤ i → i < j → j ≤ n / 2 →
      c * (Real.sqrt (n : ℝ) * (2 : ℝ) ^ i / ((i : ℝ) * Real.sqrt (i : ℝ))) ≤
        (Nat.gcd (n.choose i) (n.choose j) : ℝ) := by
…
```
Mentions: c