Erdős problem 287 · problems.science

Let $k\geq2$ . Is it true that, for any distinct integers $1 < n_1 < \cdots < n_k$ such that $\sum_{i=1}^k \frac{1}{n_i} = 1$ , we must have $\max(n_{i+1} - n_i) \geq 3$ ?

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FormalConjectures/ErdosProblems/

287.lean

Retained formal statement5 of 5

The conjecture erdos_287 would follow for all but at most finitely many exceptions if it were known that, for all large $N$ , there exists a prime $p \in [N, 2N]$ such that $\frac{p+1}{2}$ is also prime.

More precisely: if the prime conjecture holds, then there exists $k_0$ such that for all $k \geq k_0$ , any Egyptian fraction representation of $1$ with $k$ terms and all terms $> 1$ must have $\max(n_{i+1} - n_i) \geq 3$ .

FormalConjectures/ErdosProblems/287.leanErdos287.erdos_287.variants.prime_conjecture_implies5 linesExact file

1(True ↔ ∃ N₀, ∀ (N : ℕ), N₀ ≤ N → ∃ p, Nat.Prime p ∧ N ≤ p ∧ p ≤ 2 * N ∧ Nat.Prime ((p + 1) / 2)) →2  ∃ k₀,3    ∀ (k : ℕ),4      k₀ ≤ k →5        ∀ (hk : 2 ≤ k) (s : Fin k → ℕ), StrictMono s → 1 < s ⟨0, ⋯⟩ → ∑ i, 1 / ↑(s i) = 1 → 3 ≤ Erdos287.max_gap k s

TextbookStatement only, no proof

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