Every real number has a continued fraction: x = a₀ + 1/(a₁ + 1/(a₂ + ⋯)). The integers a₁, a₂, a₃, … are the partial quotients. For π they are 3; 7, 15, 1, 292, 1, 1, 1, 2… For √2 they are 1; 2, 2, 2, 2, 2… (periodic, all 2s). Khinchin proved in 1934 that for almost every real number, the geometric mean of the partial quotients converges to the same constant K₀ ≈ 2.68545.
P(k) = log₂(1 + 1/k(k+2)). The partial quotient 1 appears in ~41% of all continued fraction expansions of random real numbers.
The formula for K₀ is K₀ = ∏(k=1 to ∞) (1 + 1/(k(k+2)))^(log₂(k)), which converges extremely slowly. Khinchin's theorem is an example of a result that is true for almost every number yet cannot be verified for a single specific constant. We cannot exhibit one confirmed instance of a number obeying it.
By k=3 over two-thirds of all partial quotients are accounted for. The sequence converges slowly toward 1.
The fact that 1 dominates (41.5%) explains why K₀ ≈ 2.685 is less than 3: the small values pull the geometric mean down. If all digits from 1 to 9 were equally likely, the geometric mean would be (1·2·3⋯9)^(1/9) = 9!^(1/9) ≈ 4.15. The heavy weighting toward 1 makes K₀ considerably smaller.
Khinchin's constant K0 ≈ 2.68545 is a universal limit: for almost every real number x = [a0; a1, a2, ...], the geometric mean of the partial quotients (a1*a2*...*an)^(1/n) converges to K0. Proved by Khinchin in 1934. The striking aspect is universality: almost every number shares this geometric mean, yet the result cannot be verified for any single known constant like pi or e. Whether K0 is algebraic or transcendental is unknown.