Khinchin

Khinchin's constant $K_0 = 2.68545\ldots$ , the almost-sure geometric mean of continued-fraction terms.

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KhinchinKhinchin's constant, a new mathematical-constant symbol.

Typed signatures

Type	Package	Applies
`untyped`	`analytic`	always

Domain: Special functions

Details

For Lebesgue-almost every real number, the geometric mean of the terms $a_1, a_2, \dots$ in its continued-fraction expansion $x = [a_0; a_1, a_2, \dots]$ converges to $K_0$ , independent of $x$ -- a fact with no known elementary proof.
A symbol, like ConstGlaisher: prints as itself under plain evaluation, and resolves to a decimal only under N().

Examples

1 more kept as data, checked against the oracles

Basic #

Khinchin's constant

N(\mathrm{Khinchin})

Out

2.685\,452\,001\,065\,306\,2

Khinchin's constant to 50 digits

N(\mathrm{Khinchin}, 50)

Out

2.685\,452\,001\,065\,306\,445\,309\,714\,835\,481\,795\,693\,820\,382\,293\,994\,5

Scope #

A hundredth power rounds to its exact 43-digit integer

\mathrm{round}(\mathrm{Khinchin}^{100})

Out

7\,975\,160\,530\,073\,655\,774\,671\,985\,794\,452\,796\,861\,418\,592

Properties #

The constant lies between 2 and 3

2\lt\mathrm{Khinchin}\lt3

Out

\mathrm{True}

About

Examples	5
Checked against	Wolfram Language 4 agreeSageMath 4 agreeJulia 4 agreempmath 3 agree
Crosswalk	Wolfram Language

primitive · numericirreducibly numeric — it evaluates rather than rewrites

nativeenginetypescriptpackages/symbols/analysis/analytic/src/khinchin.ts