Khinchin's constant

In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers x, coefficients a_i of the continued fraction expansion of x have a finite geometric mean that is independent of the value of x and is known as Khinchin's constant.

That is, for

it is almost always true that

where ${\displaystyle K_{0}}$ is Khinchin's constant

(with ${\displaystyle \prod }$ denoting the product over all sequence terms).

Although almost all numbers satisfy this property, it has not been proven for any real number not specifically constructed for the purpose. Among the numbers x whose continued fraction expansions are known not to have this property are rational numbers, roots of quadratic equations (including the square roots of integers and the golden ratio Φ), and the base of the natural logarithm e.

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