387.lean`

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

5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 387

References:

erdosproblems.com/387
[ErGr76b] Erdős, P. and Graham, R. L., On the prime factors of ${n \choose k}$. Fibonacci Quart. (1976), 348-352.
[Er78g] Erdős, Pál, On prime factors of binomial coefficients. II. Mat. Lapok (1978/82), 307-316.
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[Sc58] Schinzel, A., Sur un problème de P. Erdős. Colloq. Math. (1958), 198-204.
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
[Fa66] Faulkner, M. "On a theorem of Sylvester and Schur." Journal of the London Mathematical Society 1.1 (1966): 107-110.
[BNPZ26] Bui, H., Naprienko, S., Pratt, K., and Zaharescu, A. Binomial coefficients with divisors avoiding an interval. arXiv:2605.21221 (2026).

Imports (1)

FormalConjecturesUtil

theorem Erdos387.erdos_387 Is there an absolute constant $c > 0$ such that, for all $1 \leq k < n$, the binomial coefficient $\binom{n}{k}$ has a divisor in $(cn, n]$? sorry line 52 source
Source documentation
Is there an absolute constant $c > 0$ such that, for all $1 \leq k < n$, the binomial coefficient $\binom{n}{k}$ has a divisor in $(cn, n]$?
Bui, Naprienko, Pratt, and Zaharescu [BNPZ26] answered this negatively.

depends on sorry what this means
Statement uses: Iff, False, Exists, Real, And, LT.lt, OfNat.ofNat, Nat, LE.le, Membership.mem, Set, Set.Ioc, HMul.hMul, Dvd.dvd, Nat.choose
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_387 : answer(False) ↔ ∃ c : ℝ, 0 < c ∧ ∀ n k : ℕ, 1 ≤ k → k < n →
    ∃ d : ℕ, (d : ℝ) ∈ Set.Ioc (c * n) n ∧ d ∣ n.choose k := by
…
```
Mentions: c
example Erdos387.(example at line 57) line 57 source
theorem Erdos387.erdos_387.variants.schinzel The following is Schinzel's conjecture, which appears in [Gu04] sorry line 62 source
Source documentation
The following is Schinzel's conjecture, which appears in [Gu04].

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, Not, IsPrimePow, Exists, LT.lt, Dvd.dvd, HSub.hSub, Nat.choose, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_387.variants.schinzel : answer(sorry) ↔
    ∀ᶠ k in atTop, ¬ IsPrimePow k → ∃ n : ℕ, ∀ i < k, ¬ n - i ∣ n.choose k := by
…
```
theorem Erdos387.erdos_387.variants.easy It is easy to see that n.choose k has a divisor in [n / k, n] line 68 source
Source documentation
It is easy to see that n.choose k has a divisor in [n / k, n].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, And, Membership.mem, Real, Set, Set.Icc, HDiv.hDiv, Dvd.dvd, Nat.choose
Proof uses: dite, Eq, of_eq_true, Eq.trans, True, congrArg, funext, congr, congrFun', CharP.cast_eq_zero, div_zero, true_and, Nat.choose_zero_right, eq_true, Not, Eq.mpr, id, Exists.intro, Nat.gcd, And.intro and 76 more
157 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_387.variants.easy {n : ℕ} {k : ℕ} (hn : 1 ≤ n) (hk : k ≤ n) : ∃ d : ℕ,
    (d : ℝ) ∈ Set.Icc (n / k : ℝ) n ∧ d ∣ n.choose k := by
…
```
theorem Erdos387.erdos_387.variants.guy Is it true for any $c < 1$ and all $n$ sufficiently large, for all $1 \leq k < n$, $\binom{n}{k}$ has a divisor in $(cn, n]$? sorry line 88 source
Source documentation
Is it true for any $c < 1$ and all $n$ sufficiently large, for all $1 \leq k < n$, $\binom{n}{k}$ has a divisor in $(cn, n]$?
This variant appears in [Gu04]. Bui, Naprienko, Pratt, and Zaharescu [BNPZ26] answered it negatively.

depends on sorry what this means
Statement uses: Iff, False, Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, Exists, And, Membership.mem, Set, Set.Ioc, HMul.hMul, Dvd.dvd, Nat.choose, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_387.variants.guy : answer(False) ↔ ∀ c : ℝ, c < 1 → ∀ᶠ n : ℕ in atTop, ∀ k : ℕ, 1 ≤ k →
    k < n → ∃ d : ℕ, (d : ℝ) ∈ Set.Ioc (c * n) n ∧ d ∣ n.choose k := by
…
```
Mentions: c