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 $\gamma = 0.577 \ldots$ and $\exp (\gamma ) = 1.781 \ldots .$ A preliminary step was the calculation of $\gamma$ and $\exp (\gamma )$ to 20,700D. It follows from the continued fractions that, if $\gamma$ or $\exp (\gamma )$ is of the form $P/Q$ for integers P and Q, then $|Q| > {10^{10000}}$.