# Khinchin

is Khinchin's constant, with numerical value .

# Details

• Mathematical constant treated as numeric by NumericQ and as a constant by D.
• Khinchin can be evaluated to any numerical precision using N.
• Khinchin's constant (sometimes called Khintchine's constant) is given by .

# Background & Context

• Khinchin is the symbol representing Khinchin's constant , also known as Khintchine's constant. Khinchine is defined as the limiting value for the geometric mean of the terms of a simple continued fraction expansion of a real number , where the value of is independent of the choice of . Khinchin has a numerical value and a closed form product is given by . Khinchin arises most commonly in the theory of continued fractions and in ergodic theory.
• When Khinchin is used as a symbol, it is propagated as an exact quantity.
• It is not currently known if Khinchin is rational (meaning it can be expressed as a ratio of integers), algebraic (meaning it is the root of some integer polynomial), or normal (meaning the digits in its base- expansion are equally distributed) to any base.
• Khinchin can be numerically evaluated using N. However, no efficient formulas for computing large numbers of its digits are currently known. RealDigits can be used to return a list of digits of Khinchin and ContinuedFraction to obtain terms of its continued fraction expansion.

# Examples

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## Basic Examples(1)

Evaluate to any precision:

## Scope(2)

Do an exact numerical computation:

## Applications(1)

Geometric mean of the first 1000 continued fraction terms in π:

## Properties & Relations(2)

Various symbolic relations are automatically used:

Various products give results that can be expressed using Khinchin:

## Neat Examples(1)

Terms in the continued fraction:

Wolfram Research (1999), Khinchin, Wolfram Language function, https://reference.wolfram.com/language/ref/Khinchin.html (updated 2007).

#### Text

Wolfram Research (1999), Khinchin, Wolfram Language function, https://reference.wolfram.com/language/ref/Khinchin.html (updated 2007).

#### CMS

Wolfram Language. 1999. "Khinchin." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/Khinchin.html.

#### APA

Wolfram Language. (1999). Khinchin. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Khinchin.html

#### BibTeX

@misc{reference.wolfram_2024_khinchin, author="Wolfram Research", title="{Khinchin}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/Khinchin.html}", note=[Accessed: 21-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_khinchin, organization={Wolfram Research}, title={Khinchin}, year={2007}, url={https://reference.wolfram.com/language/ref/Khinchin.html}, note=[Accessed: 21-July-2024 ]}