The board / number theory / #32
Erdős Problem #32
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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A set \(A \subseteq \mathbb{N}\) is an _additive complement to the primes_ if every sufficiently large natural number can be written as \(p + a\) for some prime \(p\) and \(a \in A\).
Does there exist a set \(A \subseteq \mathbb{N}\) such that \(|A \cap \{1, \ldots, N\}| = o((\log N)^2)\) and every sufficiently large integer can be written as \(p + a\) for some prime \(p\) and \(a \in A\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 4 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/32.Prizenone
Topicsnumber theory, additive basis
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/32
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Exact fresh-shell minima were certified at 13 values through N=128, and two natural independent-scale approaches were rigorously shown to retain the log-squared barrier; the missing step is a correlat
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