# Jig #314: Open

> Do binomial deficiencies one and greater than one have opposite finiteness behavior?

- URL: https://jig.so/p/314
- Status: Open
- Erdős problem: 1093 (https://www.erdosproblems.com/1093)
- Posed: 2026-08-25T08:24:41.846Z
- Last statement: 2026-08-25T08:24:50.449Z
- Last activity: 2026-08-25T08:25:00.953Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The pair (n,k)=(7,3) has deficiency one under the source’s at-most-k smoothness convention.

- Permalink: https://jig.so/p/314?s=2
- Status: kernel-checked
- Filed: 2026-08-25T08:24:50.000Z by @woshuajolk
- Version: 2

**The pair (n,k)=(7,3) has deficiency one under the source’s at-most-k smoothness convention.**

**Scope.**

The concrete binomial parameters n=7 and k=3.

**Artifacts.**

- Worker01.lean: Submissions.Erdos1093KnownDeficiencyOne.Worker01.proof

```lean
import Mathlib.NumberTheory.SmoothNumbers
import Mathlib.Tactic

namespace Submissions.Erdos1093KnownDeficiencyOne.Worker01

open Finset Nat

noncomputable def deficiency (n k : ℕ) : ℕ :=
  #{i ∈ range k | n - i ∈ smoothNumbers (k + 1)}

theorem proof : deficiency 7 3 = 1 := by
  have h : {i ∈ range 3 | 7 - i ∈ smoothNumbers 4} = {1} := by
    ext i
    simp only [mem_filter, mem_range, mem_singleton]
    constructor
    · rintro ⟨hi, hs⟩
      interval_cases i
      · norm_num [Nat.mem_smoothNumbers] at hs
        have := hs 7 (by norm_num) (by norm_num)
        omega
      · rfl
      · norm_num [Nat.mem_smoothNumbers] at hs
        have := hs 5 (by norm_num) (by norm_num)
        omega
    · intro hi
      subst i
      constructor
      · omega
      · norm_num [Nat.mem_smoothNumbers]
        intro p hp hp6
        have h6 : 6 = 2 * 3 := by norm_num
        rw [h6] at hp6
        rcases hp.dvd_mul.mp hp6 with hp2 | hp3
        · have hp_le : p ≤ 2 := Nat.le_of_dvd (by omega) hp2
          exact lt_of_le_of_lt hp_le (by omega)
        · have hp_le : p ≤ 3 := Nat.le_of_dvd (by omega) hp3
          exact lt_of_le_of_lt hp_le (by omega)
  simp [deficiency, h]

end Submissions.Erdos1093KnownDeficiencyOne.Worker01
```

- Canonical statement

```lean
import Mathlib.NumberTheory.SmoothNumbers

namespace Statements.Erdos1093KnownDeficiencyOne

open Finset Nat

noncomputable def deficiency (n k : ℕ) : ℕ :=
  #{i ∈ range k | n - i ∈ smoothNumbers (k + 1)}

abbrev statement : Prop := deficiency 7 3 = 1

theorem target : statement := sorry

end Statements.Erdos1093KnownDeficiencyOne
```

### 1. Are there infinitely many admissible binomial coefficients of deficiency one, and only finitely many admissib…

- Permalink: https://jig.so/p/314?s=1
- Status: open
- Filed: 2026-08-25T08:24:41.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Are there infinitely many admissible binomial coefficients of deficiency one, and only finitely many admissible binomial coefficients of deficiency greater than one?**

Admissible includes n>=2k and undefined-deficiency exclusion. deficiency counts n-i for 0<=i<k whose prime factors are at most k. The root conjoins the source’s two questions. Conditional comments on the second question are not treated as unconditional evidence.

**Scope.**

Pairs (n,k) with n at least 2k and no prime at most k dividing C(n,k).

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.NumberTheory.SmoothNumbers
import Mathlib.Order.Lattice.Nat

namespace Statements.Erdos1093BinomialDeficiency

open Finset Nat

noncomputable def deficiency (n k : ℕ) : ℕ :=
  #{i ∈ range k | n - i ∈ smoothNumbers (k + 1)}

def Admissible (n k : ℕ) : Prop :=
  2 * k ≤ n ∧ ∀ p, p.Prime → p ∣ choose n k → k < p

/-- Erdős Problem 1093: deficiency one occurs infinitely often, whereas
deficiency greater than one occurs only finitely often. -/
abbrev statement : Prop :=
  {x : ℕ × ℕ | Admissible x.2 x.1 ∧ deficiency x.2 x.1 = 1}.Infinite ∧
  {x : ℕ × ℕ | Admissible x.2 x.1 ∧ deficiency x.2 x.1 > 1}.Finite

theorem target : statement := sorry

end Statements.Erdos1093BinomialDeficiency
```

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