The Erdos conjecture on arithmetic progressions, often called the Erdos-Turan conjecture, is a conjecture in arithmetic combinatorics stating that if the sum of the reciprocals of the members of a set of positive integers diverges, then that set contains arithmetic progressions of every finite length.
Facts
StatementIf A is a set of positive integers such that the sum of the reciprocals of the members of A diverges, then A contains arithmetic progressions of every finite length. 1 Prize StatusPaul Erdos offered a cash prize for a proof of the conjecture: 3000 US dollars beginning in 1976, later raised to 5000 US dollars in 1996; the conjecture remains open and the prize unclaimed. 1 Progress Toward ResolutionBloom and Sisask improved bounds toward the related density result in 2020, and in 2023 Kelley and Meka found a substantially improved bound, which Bloom and Sisask then further improved; the full conjecture on arithmetic progressions remains unsolved. 1 Classification
Resolution Status Prize Status
Prize Status (category)Prize Offered, Unclaimed 1 Connections
In Branch
Source Erdos Conjecture on Arithmetic Progressions (Wikipedia)
Posed By
Sources
1. Erdos Conjecture on Arithmetic Progressions (Wikipedia)
Statement section
If the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions.
Prize section
Erdős offered a prize of US$3000 for a proof of this conjecture
Recent progress section
a new bound of exp(-c(log N)^(1/12))N was found by computer scientists Kelley and Meka
- In Branch: Combinatorics, Lead sentence
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