Computation of the regular continued fraction for Euler’s constant

Abstract

We describe a computation of the first 20,000 partial quotients in the regular continued fractions for Euler’s constant γ = 0.577 … \gamma = 0.577 \ldots and exp ⁡ ( γ ) = 1.781 … . \exp (\gamma ) = 1.781 \ldots . A preliminary step was the calculation of γ \gamma and exp ⁡ ( γ ) \exp (\gamma ) to 20,700D. It follows from the continued fractions that, if γ \gamma or exp ⁡ ( γ ) \exp (\gamma ) is of the form P / Q P/Q for integers P and Q, then | Q | > 10 10000 |Q| > {10^{10000}} .