384.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 384

References:

erdosproblems.com/384
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[Ec69] Ecklund, Jr., E. F., On prime divisors of the binomial coefficient. Pacific J. Math. (1969), 267-270.
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.

Imports (1)

FormalConjecturesUtil

theorem Erdos384.erdos_384 If $1k^2$ the binomial coefficient $\binom{n}{k}$ is divisible by a prime $p<n/k$ sorry line 52 source
Source documentation
If $1<k<n-1$ then $\binom{n}{k}$ is divisible by a prime $p\leq n/2$ (except $\binom{7}{3}=\binom{7}{4}=5\cdot 7$).
A conjecture of Erdős and Selfridge [ErGr80]. Proved by Ecklund [Ec69], who made the stronger conjecture that whenever $n>k^2$ the binomial coefficient $\binom{n}{k}$ is divisible by a prime $p<n/k$. Discussed in problem B31 and B33 of Guy's collection [Gu04]. Stronger forms of this conjecture are 1094 and 1095.
Ecklund's theorem is stated for $n \geq 2k$ with the single exception $\binom{7}{3}$; by the symmetry $\binom{n}{k} = \binom{n}{n-k}$ this is the form above, where $\binom{7}{4}$ is the second exception. The prime bound is $p \leq n/2$ (that is, $2p \leq n$); see erdos_384.variants.strict for the strict inequality.

L4DEL explanation
If $1k^2$ the binomial coefficient $\binom{n}{k}$ is divisible by a prime $p<n/k$. Discussed in problem B31 and B33 of Guy's collection [Gu04]. Stronger forms of this conjecture are 1094 and 1095. Ecklund's theorem is stated for $n \geq 2k$ with the single exception $\binom{7}{3}$; by the symmetry $\binom{n}{k} = \binom{n}{n-k}$ this is the form above, where $\binom{7}{4}$ is the second exception. The prime bound is $p \leq n/2$ (that is, $2p \leq n$); see erdos_384.variants.strict for the strict inequality.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, HSub.hSub, Ne, Prod, Prod.mk, Exists, And, Nat.Prime, Dvd.dvd, Nat.choose, LE.le, HMul.hMul
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_384 (n k : ℕ) (hk : 1 < k) (hkn : k < n - 1) (h₃ : (n, k) ≠ (7, 3))
    (h₄ : (n, k) ≠ (7, 4)) :
    ∃ p : ℕ, p.Prime ∧ p ∣ n.choose k ∧ 2 * p ≤ n := by
…
```
theorem Erdos384.erdos_384.variants.strict sorry line 63 source
depends on sorry what this means
Statement uses: Iff, False, Nat, LT.lt, OfNat.ofNat, HSub.hSub, Ne, Prod, Prod.mk, Exists, And, Nat.Prime, Dvd.dvd, Nat.choose, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.