683.lean`

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

5 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 683

References:

erdosproblems.com/683
[Er34] Erdős, Paul, A Theorem of Sylvester and Schur. J. London Math. Soc. (1934), 282--288.
[Er55d] Erdős, P., On consecutive integers. Nieuw Arch. Wisk. (3) (1955), 124--128.
[Er79d] Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.

Imports (2)

def Erdos683.P Let $P(n, k)$ be the largest prime factor of $\binom{n}{k}$ line 41 source
Source documentation
Let $P(n, k)$ be the largest prime factor of $\binom{n}{k}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sup, Nat.primeFactors, Nat.choose, id
3 instances, coercions and auxiliary constants left out.
```
def P (n k : ℕ) : ℕ := (n.choose k).primeFactors.sup id
…
```
theorem Erdos683.erdos_683 Let $P(n, k)$ be the largest prime factor of $\binom{n}{k}$ sorry line 54 source
Source documentation
Let $P(n, k)$ be the largest prime factor of $\binom{n}{k}$. There exists $c > 0$ such that $P(n, k) \ge \min(n - k + 1, k^{1 + c})$ for all $0 < k \le n/2$.
Erdős stated this for $1 \le k \le n$ with the bound $\min(n-k+1, k^{1+c})$ [Er79d]. The minimum is needed even for $k \le n/2$: at $n = 2k$ every prime factor of $\binom{2k}{k}$ is at most $2k$, so $P(n, k) \ge k^{1+c}$ fails for large $k$. The range $k \le n/2$ is natural (cf. #961 and the discussion).

L4DEL explanation
Let $P(n, k)$ be the largest prime factor of $\binom{n}{k}$. There exists $c > 0$ such that $P(n, k) \ge \min(n - k + 1, k^{1 + c})$ for all $0 < k \le n/2$. Erdős stated this for $1 \le k \le n$ with the bound $\min(n-k+1, k^{1+c})$ [Er79d]. The minimum is needed even for $k \le n/2$: at $n = 2k$ every prime factor of $\binom{2k}{k}$ is at most $2k$, so $P(n, k) \ge k^{1+c}$ fails for large $k$. The range $k \le n/2$ is natural (cf. #961 and the discussion).
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Nat, LT.lt, LE.le, HDiv.hDiv, GE.ge, Erdos683.P, Min.min, HAdd.hAdd, HSub.hSub, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_683 : answer(sorry) ↔
    ∃ c > (0 : ℝ), ∀ n k : ℕ, 0 < k ∧ k ≤ n / 2 →
      (P n k : ℝ) ≥ min (↑(n - k + 1) : ℝ) ((k : ℝ) ^ (1 + c)) := by
…
```
Mentions: c, Erdos683.P
theorem Erdos683.erdos_683.variant.sylvester_schur Sylvester and Schur [Er34] proved that $P(n, k) > k$ for $k \le n/2$ sorry line 63 source
Source documentation
Sylvester and Schur [Er34] proved that $P(n, k) > k$ for $k \le n/2$.

depends on sorry what this means
Statement uses: Nat, And, LT.lt, OfNat.ofNat, LE.le, HDiv.hDiv, GT.gt, Erdos683.P
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_683.variant.sylvester_schur :
    ∀ n k : ℕ, 0 < k ∧ k ≤ n / 2 → P n k > k := by
…
```
Mentions: Erdos683.P
theorem Erdos683.erdos_683.variant.erdos_log Erdős [Er55d] improved this to $P(n, k) \gg \min(n - k + 1, k \log k)$ for $k \le n/2$ sorry line 72 source
Source documentation
Erdős [Er55d] improved this to $P(n, k) \gg \min(n - k + 1, k \log k)$ for $k \le n/2$. The minimum cannot be dropped: at $n = 2k$ one has $P(n, k) \le 2k$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Nat, LT.lt, LE.le, HDiv.hDiv, GE.ge, Erdos683.P, Min.min, HAdd.hAdd, HSub.hSub, HMul.hMul, Real.log
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_683.variant.erdos_log :
    ∃ c > 0, ∀ n k : ℕ, 0 < k ∧ k ≤ n / 2 →
      (P n k : ℝ) ≥ min (↑(n - k + 1) : ℝ) (c * k * Real.log k) := by
…
```
Mentions: c, Erdos683.P
theorem Erdos683.erdos_683.variant.exp_sqrt Standard heuristics suggest that $P(n, k) > \min(n - k + 1, e^{c\sqrt{k}})$ for some constant $c > 0$ sorry line 82 source
Source documentation
Standard heuristics suggest that $P(n, k) > \min(n - k + 1, e^{c\sqrt{k}})$ for some constant $c > 0$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Nat, LT.lt, LE.le, HDiv.hDiv, Erdos683.P, Min.min, HAdd.hAdd, HSub.hSub, Real.exp, HMul.hMul, Real.sqrt
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_683.variant.exp_sqrt :
    ∃ c > 0, ∀ n k : ℕ, 0 < k ∧ k ≤ n / 2 →
      (P n k : ℝ) > min (↑(n - k + 1) : ℝ) (Real.exp (c * Real.sqrt k)) := by
…
```
Mentions: c, Erdos683.P