Erdős conjecture on arithmetic progressions

Open

erdos-arithmetic-progressions

Statement

A \subseteq \mathbb{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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