1. FormalConjectures
  2. ErdosProblems
  3. «730»
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Erdős Problem 730

References:

namespace Erdos730 abbrev S := {(n, m) : ℕ × ℕ | n < m ∧ n.centralBinom.primeFactors = m.centralBinom.primeFactors}

Are there infinitely many pairs of integers $n < m$ such that $\binom{2n}{n}$ and $\binom{2m}{m}$ have the same set of prime divisors?

@[category research open, AMS 11] theorem declaration uses 'sorry'erdos_730 : answer(sorry) ↔ S.Infinite := ⊢ True ↔ S.Infinite All goals completed! 🐙

For example, $(87,88)$ and $(607,608)$ are such pairs.

@[category textbook, AMS 11] theorem erdos_730.variants.explicit_pairs : {(87, 88), (607, 608)} ⊆ S := ⊢ {(87, 88), (607, 608)} ⊆ S ⊢ (87, 88) ∈ S⊢ (607, 608) ∈ S ⊢ (87, 88) ∈ S⊢ (607, 608) ∈ S exact ⟨⊢ 607 < 608 All goals completed! 🐙, ⊢ (Nat.centralBinom 607).primeFactors = (Nat.centralBinom 608).primeFactors All goals completed! 🐙⟩

There are examples where $(n, m) ∈ S$ with $m ≠ n + 1$.

(Found by AlphaProof, although it was implicit already in [A129515])

@[category research solved, AMS 11] theorem erdos_730.variants.delta_ne_one : ∃ (n m : ℕ), (n, m) ∈ S ∧ m ≠ n + 1 := ⊢ ∃ n m, (n, m) ∈ S ∧ m ≠ n + 1 ⊢ ∃ n m, (n < m ∧ n.centralBinom.primeFactors = m.centralBinom.primeFactors) ∧ ¬m = n + 1 ⊢ ∃ m, (10003 < m ∧ (Nat.centralBinom 10003).primeFactors = m.centralBinom.primeFactors) ∧ ¬m = 10003 + 1 ⊢ (10003 < 10005 ∧ (Nat.centralBinom 10003).primeFactors = (Nat.centralBinom 10005).primeFactors) ∧ ¬10005 = 10003 + 1 ⊢ ∀ (a : ℕ), Nat.Prime a → (a ∣ Nat.choose 20006 10003 ↔ a ∣ Nat.choose 20010 10005) simp_rw ⊢ ∀ (a : ℕ), Nat.Prime a → (a ∣ Nat.choose 20006 10003 ↔ a ∣ Nat.choose 20010 10005)Nat.choose_eq_descFactorial_div_factorial] intro p p:ℕhp:Nat.Prime p⊢ p ∣ Nat.descFactorial 20006 10003 / Nat.factorial 10003 ↔ p ∣ Nat.descFactorial 20010 10005 / Nat.factorial 10005 p:ℕhp:Nat.Prime p⊢ p ∣ Nat.descFactorial 20006 10003 / Nat.factorial 10003 → p ∣ Nat.descFactorial 20010 10005 / Nat.factorial 10005p:ℕhp:Nat.Prime p⊢ p ∣ Nat.descFactorial 20010 10005 / Nat.factorial 10005 → p ∣ Nat.descFactorial 20006 10003 / Nat.factorial 10003 all_goals exact fun h' => or_self_iff.1 (hp.dvd_mul.1 ( h'.trans (p:ℕhp:Nat.Prime ph':p ∣ Nat.descFactorial 20010 10005 / Nat.factorial 10005⊢ Nat.descFactorial 20010 10005 / Nat.factorial 10005 ∣ Nat.descFactorial 20006 10003 / Nat.factorial 10003 * (Nat.descFactorial 20006 10003 / Nat.factorial 10003) refine' of_decide_eq_true (p:ℕhp:Nat.Prime ph':p ∣ Nat.descFactorial 20010 10005 / Nat.factorial 10005⊢ decide (Nat.descFactorial 20010 10005 / Nat.factorial 10005 ∣ Nat.descFactorial 20006 10003 / Nat.factorial 10003 * (Nat.descFactorial 20006 10003 / Nat.factorial 10003)) = ?m.96 All goals completed! 🐙 : _ = ↑_)))) end Erdos730