Fast Algorithms for High-Precision Computation of Elementary Functions

Open problem: Is γ or exp(γ) rational ? Since the regular continued fraction gives best rational approximations, continued fraction computations can give theorems of the form: If x is rational, say x = p/q, then |q| > B for some (very large) bound B. To obtain a result like this with given bound B, we need to compute x with absolute error O(1/B2). Using this method we know that, if γ or exp(γ) is rational, say p/q, then |q| must be very large. Thus, in some sense these numbers are “unlikely” to be rational.