Every real number can be written as a continued fraction: x = a₀ + 1/(a₁ + 1/(a₂ + ⋯)) where the aᵢ are positive integers. For π: a₀=3, a₁=7, a₂=15, a₃=1, a₄=292, a₅=1… For e: a₀=2, a₁=1, a₂=2, a₃=1, a₄=1, a₅=4…
In 1934, Aleksandr Khinchin proved: for almost every real number x, the geometric mean (a₁ · a₂ · a₃ ⋯ aₙ)^(1/n) converges to K₀ ≈ 2.685 as n→∞. "Almost every" has a precise meaning: the exceptions: rational numbers, quadratic irrationals like φ: form a set of measure zero.
Remarkably, no specific example of a "normal" number (one that actually achieves K₀) has been proved. Whether π, e, or √2 satisfy Khinchin's theorem is unknown. We know the theorem holds for almost all reals: but cannot exhibit a single confirmed instance.