Erdős conjecture on arithmetic progressions

Open

erdos-arithmetic-progressions

Statement

If $A \subseteq N$ has divergent reciprocal sum $\sum_{a \in A} 1/ a = \infty$ , prove it contains arbitrarily long arithmetic progressions.

Current frontier

erdosproblems.com/3, OPEN. The 3-term case is settled (Bloom–Sisask 2020); Szemerédi (1975) handles positive density and Green–Tao (2008) the primes. The conjecture for $k \geq 4$ under the divergence hypothesis is open.

When this counts as solved

OPEN-COMPLETION

PROOF_COMPLETE for all $k$ , COUNTEREXAMPLE for a divergent-reciprocal set missing some $k$ -term progression, or BREAKTHROUGH for settling a new fixed $k \geq 4$ (the $k = 3$ case is already done).

Classification

Open-completion

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