khinchin's constant

In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers x, the infinitely many terms a_i of the continued fraction expansion of x have an astonishing property: their geometric mean is a constant, known as Khinchin's constant, which is independent of the value of x. That is, for

x = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + ...}}}

it is almost always true that

\lim_{n \rightarrow \infty } \left( \prod_{i=1}^n a_i \right) ^{1/n} = K = \prod_{r=1}^\infty {\left\{ 1+{1\over r(r+2)}\right\}}^{\log_2 r}  \approx 2.6854520010\dots

Among the numbers x whose continued fraction expansions do not have this property are rational numbers, solutions of quadratic equations with rational coefficients (including the golden ratio φ), and the base of the natural logarithms e. Among the numbers whose continued fraction expansions apparently do have this property (based on numerical evidence) are π, the Euler-Mascheroni constant γ, and Khinchin's constant itself. However this is unproven, because even though almost all real numbers are known to have this property, it has not been proven for any specific real number. Khinchin is sometimes spelled Khintchine (the French transliteration) in older mathematical literature.

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