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Source. The paragraph on p. 44 of P. Erdős, Combinatorial problems in geometry, Math. Chronicle 12 (1983), 35--54, the transcript of an invited address at the 17th New Zealand Mathematics Colloquium (Dunedin, 17--19 May 1982), as named on the source card. The lecture numbers none of its statements; the pages are the journal's own.

Statement

Conjecture (p. 44). If $a_1<a_2<\cdots$ is an infinite sequence of integers with $\sum 1/a_i=\infty$, then the sequence contains arbitrarily long arithmetic progressions; in the lecture's second wording, for every $k$ there are terms $a_{i_1},\dots,a_{i_k}$ forming an arithmetic progression of $k$ terms.

Erdős says he conjectured this more than forty years before, offers a prize for it, and notes that by Euler's divergence of the sum of the reciprocals of the primes it would imply arbitrarily long arithmetic progressions among the primes. He recalls as context Szemerédi's theorem (1972), that a sequence of positive density contains arbitrarily long arithmetic progressions, and Furstenberg's ergodic proof.

Read depth. Claims checked: the passage was read clause by clause on the page images of the print. A second reader checked the statement, hypotheses, label and page against the print.

Proof pointer

A conjecture; the paper treats it as open.

Dependencies

None.

Bears on

• Problem 3: the conjecture is the problem's question, stated in the affirmative with the prize offer. The paper proves nothing toward it.