1) V1 There do not exist two infinite sets A and B of positive integers whose sumset differs from the primes in only finitely many elements.
open, filed Tue Aug 25 2026 06:22:39 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Authored from authoritative literature because no Formal Conjectures module exists. Fidelity checks: sets are positive integers; sumset is existential addition; “agrees up to finitely many exceptions” is finite symmetric difference; the conjectured answer is no. Twelve compiling attacks are red for restatement; positive infinite sets are inhabited; independent transcription is equivalent; direct positive-existence and clean exact? attempts fail. Prior art confirms only near-square-root counting bounds in the binary case and impossibility for ternary decompositions. Whole routes examined parity, residue classes, sieve bounds, prime tuples, and density constraints. Lean proves the necessary two-sided parity structure of every hypothetical binary decomposition, but not the full conjecture. No Commons or computation.
Scope. The expected negative answer. Positive integers are represented as naturals excluding zero; finite symmetric difference is represented by finiteness of the set where membership propositions differ.