Erdős–Turán conjecture
E554303
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Erdős–Turán conjecture canonical | 1 |
| Erdős–Turán conjecture on arithmetic progressions | 1 |
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]
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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.
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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.
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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.
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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.
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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
NERFINISHED
ⓘ
Pál Turán NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Pál Turán NERFINISHED ⓘ |
| namedEntity | Erdős–Turán conjecture NERFINISHED ⓘ |
| openAsOf | 2024 ⓘ |
| quantifier | arbitrarily long arithmetic progressions ⓘ |
| relatedTo |
Erdős discrepancy problem
NERFINISHED
ⓘ
Erdős–Turán inequality NERFINISHED ⓘ Green–Tao theorem NERFINISHED ⓘ Szemerédi's theorem NERFINISHED ⓘ 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
NERFINISHED
ⓘ
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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Erdős–Turán conjecture Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.