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Erdős Problem #3 — Arithmetic progressions in sets with divergent reciprocal sum

About 52 years old · traced to

I conjectured long ago that if

∑i1/ai=∞(6)\sum_i 1/a_i = \infty \tag{6}

then the aia_i's contain arbitrarily long arithmetic progressions. If true this of course implies that there are arbitrarily long arithmetic progressions all whose terms are primes. I offer 3000 dollars for a proof or disproof of (6).

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Erdős's divergent-reciprocal-sum conjecture on arithmetic progressions

    Let A⊂Z+A\subset\mathbb{Z^{+}} satisfy

    ∑n∈An−1=+∞.\sum_{n\in A}n^{-1}=+\infty.

    Erdős's conjecture. Then AA contains arbitrarily long arithmetic progressions. This generalizes the corresponding conjecture for the primes; the source does not state a resolution, so the problem remains open.

    source: Weiwen Zhang, “Roth-type theorems in additive combinatroics”, arXiv:2512.08455 (2025).

References

Primary source

P. Erdős, A survey of problems in combinatorial number theory, Ann. Discrete Math. 6 (1980), 89-115.

Additional references

P. Erdős, A survey of problems in combinatorial number theory, Ann. Discrete Math. 6 (1980), 89-115.

additive-combinatoricsarithmetic-progressionsnumber-theory
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