Geometric Mean (\(GM_n\))
2.685
Computed across partial quotients \(a_1 \dots a_n\)
Khinchin Constant (\(K_0\))
2.685452
Theoretical universal limit
Relative Delta (\(\Delta\))
-0.017%
Matches Khinchin behavior
Max Partial Quotient
292
Term \(a_4 = 292\) in π
Running Geometric Mean Convergence \(\left(\prod_{i=1}^n a_i\right)^{1/n}\)
\(GM_n\)
\(K_0 \approx 2.685\)
Quotient Distribution vs Gauss-Kuzmin Law
Empirical
\(P(k) = -\log_2(1-\frac{1}{(k+1)^2})\)
Continued Fraction Partial Quotients \([a_0; a_1, a_2, \dots, a_n]\)
Convergents Table (\(p_n / q_n\)) & Best Rational Approximations
| \(n\) | \(a_n\) | Convergent Fraction \(\frac{p_n}{q_n}\) | Decimal Value | Approximation Error | Geometric Mean \(GM_n\) |
|---|
The Magic of Khinchin's Theorem (1934)
Aleksandr Khinchin proved that for almost all real numbers (in the Lebesgue measure sense), the geometric mean of the continued fraction partial quotients \(a_1, a_2, \dots, a_n\) converges to a universal constant:
\(\lim_{n \to \infty} \left(\prod_{i=1}^n a_i\right)^{1/n} = K_0 \approx 2.6854520010653064453... = \prod_{k=1}^\infty \left(1 + \frac{1}{k(k+2)}\right)^{\log_2 k}\)
Why do certain numbers deviate?
- Quadratic Irrationals (\(\sqrt{2}, \sqrt{3}, \phi\)): Their continued fractions are periodic. For \(\sqrt{2} = [1; 2, 2, 2\dots]\), the geometric mean is trivially \(2.000\). For the Golden Ratio \(\phi = [1; 1, 1, 1\dots]\), it is exactly \(1.000\).
- Euler's Number \(e\): \(e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, \dots]\) has an arithmetic progression of even numbers, causing \(\lim GM_n \to \infty\).
- Pi (\(\pi\)) and Random Floats: Not algebraically constrained; their quotients follow the Gauss-Kuzmin distribution and faithfully hover around \(K_0 \approx 2.685\).