Erdős–Turán conjecture

E554303

mathematical conjecture problem in additive number theory unsolved problem in mathematics

The Erdős–Turán conjecture is an unsolved problem in additive number theory asserting that any subset of the positive integers with divergent sum of reciprocals must contain arbitrarily long arithmetic progressions.

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All labels observed (2)

Label	Occurrences
Erdős–Turán conjecture canonical	3
Erdős–Turán conjecture on arithmetic progressions	2

How this entity was disambiguated

This entity first appeared as the object of triple T5896710 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Erdős–Turán conjecture
Context triple: [Pál Erdős, knownFor, Erdős–Turán conjecture]

A. Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
B. Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
C. Waring's problem
Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.
D. Green–Tao theorem
The Green–Tao theorem is a landmark result in number theory proving that the sequence of prime numbers contains arbitrarily long arithmetic progressions.
E. Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Erdős–Turán conjecture
Target entity description: The Erdős–Turán conjecture is an unsolved problem in additive number theory asserting that any subset of the positive integers with divergent sum of reciprocals must contain arbitrarily long arithmetic progressions.

A. Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
B. Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
C. Waring's problem
Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.
D. Green–Tao theorem
The Green–Tao theorem is a landmark result in number theory proving that the sequence of prime numbers contains arbitrarily long arithmetic progressions.
E. Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	mathematical conjecture ⓘ problem in additive number theory ⓘ unsolved problem in mathematics ⓘ
coAuthor	Paul Erdős ⓘ linked to: Pál Erdős Pál Turán ⓘ
concerns	arithmetic progressions ⓘ divergent series of reciprocals ⓘ subsets of the positive integers ⓘ
conclusion	the subset contains arithmetic progressions of every finite length ⓘ
condition	the sum of reciprocals of the elements of the subset diverges ⓘ
contrastWith	Szemerédi's theorem, which uses combinatorial density conditions ⓘ linked to: Szemerédi's theorem
difficulty	considered very difficult ⓘ
doesNotRequire	positive asymptotic density of the subset ⓘ
domainRestriction	subsets of the positive integers ⓘ
field	additive number theory ⓘ number theory ⓘ
hasAbbreviation	Erdős–Turán conjecture on arithmetic progressions ⓘ linked to: Erdős–Turán conjecture
hasConsequence	would generalize many known results on arithmetic progressions in special sets if proved ⓘ
hasFormulation	If A is a subset of the positive integers and sum_{a in A} 1/a diverges, then A contains arithmetic progressions of every finite length. ⓘ
implies	existence of 3-term arithmetic progressions under the divergence condition ⓘ existence of arbitrarily long arithmetic progressions ⓘ existence of k-term arithmetic progressions for every positive integer k under the divergence condition ⓘ
involves	infinite subsets of the positive integers ⓘ
language	stated in the language of additive combinatorics ⓘ
motivatedBy	study of additive structure in large sets of integers ⓘ
namedAfter	Paul Erdős ⓘ linked to: Pál Erdős Pál Turán ⓘ
namedEntity	Erdős–Turán conjecture ⓘ
openAsOf	2024 ⓘ
quantifier	arbitrarily long arithmetic progressions ⓘ
relatedTo	Erdős discrepancy problem ⓘ Erdős–Turán inequality ⓘ Green–Tao theorem ⓘ Szemerédi's theorem ⓘ problems on sets of multiples and additive bases ⓘ
requires	divergence of the harmonic sum over the subset ⓘ
statementInformal	Any subset of the positive integers whose sum of reciprocals diverges contains arbitrarily long arithmetic progressions. ⓘ
status	open ⓘ
strongerThan	Green–Tao theorem for primes ⓘ linked to: Green–Tao theorem assertions about existence of only finitely long progressions ⓘ
topic	arithmetic progressions in dense sets ⓘ density conditions for arithmetic progressions ⓘ
typeOfCondition	analytic density condition ⓘ
usesConcept	arithmetic progression ⓘ divergent series ⓘ reciprocal sums ⓘ
yearProposed	1936 ⓘ

How these facts were elicited

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

Pál Erdős → knownFor → Erdős–Turán conjecture ⓘ

Green–Tao theorem → relatedTo → Erdős–Turán conjecture on arithmetic progressions ⓘ

linked to: Erdős–Turán conjecture

Erdős discrepancy problem → relatedTo → Erdős–Turán conjecture ⓘ

Erdős–Turán conjecture → namedEntity → Erdős–Turán conjecture ⓘ

Erdős–Turán conjecture → hasAbbreviation → Erdős–Turán conjecture on arithmetic progressions ⓘ