FormalConjectures.Wikipedia.Singmaster

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import FormalConjecturesUtil

Singmaster's conjecture

Singmaster's conjecture says that for any integer $t>1$, the number of solutions to the equation:

$\binom{n}{k} = t,\quad 1 \le k < n,$

with $\binom{n}{k}$ being the numbers that appear in Pascal's triangle, is bounded by a global constant $O(1)$.

Reference: Wikipedia


namespace Singmaster

The set of pairs (n, k) representing the solutions to the equation Nat.choose n k = t for a given t, under the constraint 1 ≤ k < n.

def solutions (t : ℕ) : Set (ℕ × ℕ) :=
  {(n, k) | 1 ≤ k ∧ k < n ∧ Nat.choose n k = t}

Singmaster's conjecture: the number of times any number $t > 1$ appears in Pascal's triangle is bounded.

@[category research open, AMS 11]
theorem declaration uses 'sorry'singmaster: ∃ (C : ℕ), ∀ (t : ℕ), t > 1 →
    (Singmaster.solutions t).Finite ∧ (Singmaster.solutions t).ncard ≤ C := by⊢ ∃ C, ∀ t > 1, (solutions t).Finite ∧ (solutions t).ncard ≤ C
  sorryAll goals completed! 🐙


end Singmaster