PROVED
This has been solved in the affirmative.
Given a finite set of primes $Q=Q_0$, define a sequence of sets $Q_i$ by letting $Q_{i+1}$ be $Q_i$ together with all primes formed by adding three distinct elements of $Q_i$. Is there some initial choice of $Q$ such that the $Q_i$ become arbitrarily large?
A problem of Ulam. In particular, what about $Q=\{3,5,7,11\}$?
Mrazović and Kovač, and independently Alon, have observed that the existence of some valid choice of $Q$ follows easily from Vinogradov's theorem that every large odd integer is the sum of three distinct primes. In particular, there exists some $N$ such that every prime $>N$ is the sum of three distinct (smaller) primes. We may then take $Q_0$ to be the set of all primes $\leq N$ (in which case all primes are eventually in some $Q_i$).
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Additional thanks to: Noga Alon, Rudi Mrazovic, and Vjekoslav Kovac
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #471, https://www.erdosproblems.com/471, accessed 2026-01-16
