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 | 3 |
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.
-
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
ⓘ
Kubilius model in probabilistic number theory ⓘ
linked to:
Turán–Kubilius inequality
|
| 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
ⓘ
Paul Erdős ⓘ
linked to:
Pál Erdős
|
| normalization | (ω(n) − log log n) / sqrt(log log n) ⓘ |
| normalOrder | ω(n) has normal order log log n ⓘ |
| originalAuthors |
Mark Kac
ⓘ
Paul Erdős ⓘ
linked to:
Pál Erdős
|
| 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
ⓘ
linked to:
Hardy–Ramanujan asymptotic formula
Turán–Kubilius inequality ⓘ 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
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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 (3)
Full triples — surface form annotated when it differs from this entity's canonical label.