Erdős Problem 493

References:

• erdosproblems.com/493
• [Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.

theorem Erdos493.erdos_493 : True ↔ ∃ (k : ℕ) (N : ℤ), ∀ (n : ℤ), N ≤ n → ∃ (a : Fin k → ℤ), (∀ (i : Fin k), 2 ≤ a i) ∧ ∏ i : Fin k, a i - ∑ i : Fin k, a i = n

Does there exist a $k$ such that every sufficiently large integer can be written in the form [\prod_{i=1}^k a_i - \sum_{i=1}^k a_i] for some integers $a_i\geq 2$?

Erdős attributes this question to Schinzel. Eli Seamans has observed that the answer is yes (with $k=2$) for a very simple reason: $n = 2(n+2)-(2+(n+2))$. There may well have been some additional constraint in the problem as Schinzel posed it, but [Er61] does not record what this is.