Khinchin's constant
In number theory, Khinchin proved that for almost all real numbers x, the infinitely many terms ai 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
- <math>x = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + ...}}} <math>
it is almost always true that
- <math>\lim_{n \rightarrow \infty } \left( \prod_{i=1}^n a_i \right) ^{1/n} = K \approx 2.6854520010\dots<math>
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 we know that almost all real numbers 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.
