Erdős–Kac theorem
E554297
The Erdős–Kac theorem is a fundamental result in probabilistic number theory stating that the number of distinct prime factors of a typical integer behaves like a normally distributed random variable.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Erdős–Kac theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5896703 — 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–Kac theorem Context triple: [Pál Erdős, knownFor, Erdős–Kac theorem]
-
A.
Dirichlet's theorem on arithmetic progressions
Dirichlet's theorem on arithmetic progressions is a fundamental result in number theory stating that any arithmetic progression with first term and difference coprime contains infinitely many prime numbers.
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B.
prime number theorem
The prime number theorem is a fundamental result in number theory that describes how prime numbers become less frequent and provides an approximate formula for the number of primes less than a given large number.
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C.
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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D.
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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E.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
- 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–Kac theorem Target entity description: The Erdős–Kac theorem is a fundamental result in probabilistic number theory stating that the number of distinct prime factors of a typical integer behaves like a normally distributed random variable.
-
A.
Dirichlet's theorem on arithmetic progressions
Dirichlet's theorem on arithmetic progressions is a fundamental result in number theory stating that any arithmetic progression with first term and difference coprime contains infinitely many prime numbers.
-
B.
prime number theorem
The prime number theorem is a fundamental result in number theory that describes how prime numbers become less frequent and provides an approximate formula for the number of primes less than a given large number.
-
C.
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.
-
D.
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.
-
E.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
probabilistic number theory result
ⓘ
theorem in number theory ⓘ |
| appliesTo |
additive functions with suitable variance growth
ⓘ
squarefree kernel statistics ⓘ |
| assumption | integers are chosen uniformly from {1,…,x} ⓘ |
| asymptoticDistribution | normal distribution ⓘ |
| classification | result about multiplicative structure of integers ⓘ |
| concernsFunction |
Ω(n)
ⓘ
ω(n) ⓘ |
| describes | distribution of the number of distinct prime factors of integers ⓘ |
| field |
number theory
ⓘ
probabilistic number theory ⓘ |
| hasGeneralizations |
Erdős–Wintner theorem
NERFINISHED
ⓘ
Kubilius model in probabilistic number theory NERFINISHED ⓘ |
| implies | normalized count of prime factors converges in distribution to standard normal ⓘ |
| inspired | further work on probabilistic methods in number theory ⓘ |
| involves |
distribution of additive arithmetic functions
ⓘ
prime factorization of integers ⓘ |
| limitDistribution | standard normal distribution ⓘ |
| limitProcess | x tends to infinity ⓘ |
| mainStatement | the number of distinct prime factors of a typical integer behaves like a normally distributed random variable ⓘ |
| mathematicalArea |
analytic number theory
ⓘ
probability theory ⓘ |
| meanAsymptotic | log log n ⓘ |
| namedAfter |
Mark Kac
NERFINISHED
ⓘ
Paul Erdős NERFINISHED ⓘ |
| normalization | (ω(n) − log log n) / sqrt(log log n) ⓘ |
| normalOrder | ω(n) has normal order log log n ⓘ |
| originalAuthors |
Mark Kac
NERFINISHED
ⓘ
Paul Erdős NERFINISHED ⓘ |
| probabilisticInterpretation | values of arithmetic functions over integers behave like sums of independent random variables ⓘ |
| publicationYear | 1940 ⓘ |
| randomVariableModeled |
Ω(n)
ⓘ
ω(n) ⓘ |
| relatedConcept |
Hardy–Ramanujan theorem
NERFINISHED
ⓘ
Turán–Kubilius inequality NERFINISHED ⓘ central limit theorem ⓘ |
| shows | fluctuations of ω(n) around log log n are typically of size sqrt(log log n) ⓘ |
| strengthens | Hardy–Ramanujan result on normal order of ω(n) ⓘ |
| topic |
additive arithmetic functions
ⓘ
distribution of prime factors ⓘ |
| type |
central limit theorem analogue
ⓘ
limit theorem ⓘ |
| typicalIntegerRange | 1 ≤ n ≤ x with x → ∞ ⓘ |
| usedIn |
analytic number theory
ⓘ
probabilistic models of integers ⓘ |
| varianceAsymptotic | log log n ⓘ |
| yearProved | 1939 ⓘ |
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–Kac theorem Description of subject: The Erdős–Kac theorem is a fundamental result in probabilistic number theory stating that the number of distinct prime factors of a typical integer behaves like a normally distributed random variable.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.