Khinchin's constant
E378994
Khinchin's constant is a mathematical constant that arises in metric number theory, describing the almost-sure geometric mean of the partial quotients in the continued fraction expansions of real numbers.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Khinchin's constant canonical | 5 |
| Khinchin constant | 1 |
| Khinchin's theorem on continued fractions | 1 |
| Khintchine's constant | 1 |
| Khintchine’s constant | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3677821 — 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: Khinchin's constant Context triple: [Aleksandr Khinchin, notableWork, Khinchin's constant]
-
A.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
B.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
-
C.
Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
-
D.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
-
E.
Pi
Pi is a 1998 psychological thriller film directed by Darren Aronofsky that explores obsession, mathematics, and mysticism through the story of a reclusive number theorist.
- 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: Khinchin's constant Target entity description: Khinchin's constant is a mathematical constant that arises in metric number theory, describing the almost-sure geometric mean of the partial quotients in the continued fraction expansions of real numbers.
-
A.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
B.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
-
C.
Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
-
D.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
-
E.
Pi
Pi is a 1998 psychological thriller film directed by Darren Aronofsky that explores obsession, mathematics, and mysticism through the story of a reclusive number theorist.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical constant
ⓘ
real number ⓘ |
| alsoKnownAs |
Khinchin's constant
ⓘ
surface form:
Khinchin constant
Khinchin's constant ⓘ
surface form:
Khintchine's constant
|
| appearsIn |
Khintchine theorem
ⓘ
surface form:
Khinchin's theorem on continued fractions
probability theory applications to number theory ⓘ |
| approximateValue |
2.6854520010
ⓘ
2.68545200106530644530971483548179569382038229399446295305115 ⓘ |
| category |
constants in metric number theory
ⓘ
constants related to continued fractions ⓘ |
| definition | the almost-sure limit of the geometric mean of the partial quotients in the simple continued fraction expansions of real numbers ⓘ |
| discoveredBy | Aleksandr Khinchin ⓘ |
| discoveryYear | 1934 ⓘ |
| field |
continued fractions
ⓘ
metric number theory ⓘ number theory ⓘ |
| greaterThan | 2 ⓘ |
| hasDecimalExpansion | 2.68545200106530644530971483548179569382038229399446295305115… ⓘ |
| hasInfiniteProductRepresentation | K_0 = ∏_{m=1}^{∞} (1 + 1/(m(m+2)))^{ rac{1}{ ext{log}_2(m+1) - ext{log}_2 m}} (up to equivalent standard forms) ⓘ |
| isIrrational | true ⓘ |
| isPositive | true ⓘ |
| isTranscendental | unknown ⓘ |
| lessThan | 3 ⓘ |
| limitExpression | K_0 = lim_{n→∞} (a_1 a_2 … a_n)^{1/n} for almost all real numbers with continued fraction partial quotients a_k ⓘ |
| measureContext | Lebesgue measure on the real numbers ⓘ |
| namedAfter | Aleksandr Khinchin ⓘ |
| namedInLanguage | Russian ⓘ |
| occursIn |
probabilistic study of continued fractions
ⓘ
typical behavior of continued fraction digits ⓘ |
| property |
defined via an infinite product over positive integers involving the Gauss–Kuzmin distribution
ⓘ
for almost every real number x, the geometric mean of the first n partial quotients of its continued fraction converges to this constant as n tends to infinity ⓘ independent of the particular real number for almost all real numbers with respect to Lebesgue measure ⓘ |
| relatedConstant |
Gauss's constant for continued fractions
ⓘ
Khinchin–Lévy constant ⓘ
surface form:
Lévy's constant
|
| relatedTo |
Gauss measure
ⓘ
Gauss–Kuzmin distribution ⓘ ergodic theory ⓘ metric Diophantine approximation ⓘ partial quotients of continued fractions ⓘ simple continued fraction expansion of real numbers ⓘ |
| symbol | K_0 ⓘ |
| usedFor |
characterizing almost-sure behavior of continued fraction expansions
ⓘ
describing typical size of continued fraction digits ⓘ |
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: Khinchin's constant Description of subject: Khinchin's constant is a mathematical constant that arises in metric number theory, describing the almost-sure geometric mean of the partial quotients in the continued fraction expansions of real numbers.
Referenced by (9)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Khinchin's theorem on continued fractions
subject surface form:
Aleksandr Khinchin
this entity surface form:
Khintchine's constant
this entity surface form:
Khinchin constant
this entity surface form:
Khintchine’s constant