PPL 052 / 177 permanent IDsPrize Problem Ledger · PPLChecked 2026.07.26
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Prize Problem LedgerErdősPPL 052Erdős Problem #3

Permanent problem IDPPL 052

Verified openErdősconjecture

Number theory

Erdős Problem #3

If A⊆ ℕ has ∑_(n∈ A)(1/n)=∈fty then must A contain arbitrarily long arithmetic progressions?

number theoryadditive combinatoricsarithmetic progressions
01

The problem

If A⊆ ℕ has ∑_(n∈ A)(1/n)=∈fty then must A contain arbitrarily long arithmetic progressions?

Open since1974earliest source
Last checked2026.07.26Catalog verification
02

Reward offers

Offer 01$5000Paul Erdős / Combinatorics Foundation
Documented

A solution must appear in a reputable journal, with documentation that Erdős offered the displayed amount. Claims are administered by the Combinatorics Foundation; erdosproblems.com does not pay awards.

Award source ↗Rules ↗
03

Sources & reading

6 unique reference linksCanonical problem page Claim or award rules OEIS A003002 OEIS A003003 OEIS A003004 OEIS A003005
04

Related open targets

Number theoryBeal conjecturePPL 005 · $1,000,000 · 33 years openNumber theoryBirch–Swinnerton-Dyer conjecturePPL 006 · $1,000,000 · 61 years openNumber theoryRiemann hypothesisPPL 138 · $1,000,000 · 167 years open