Revisited Using Continued Fractions

If the equation of the title has an integer solution with k ≥ 2, then m > 109.3·10 6 . This was the current best result and proved using a method due to L. Moser (1953). This approach cannot be improved to reach the benchmark m > 1010 7 . Here we achieve m > 1010 9 by showing that 2k/(2m−3) is a convergent of log 2 and making an extensive continued fraction digits calculation of (log 2)/N , with N an appropriate integer. This method is very different from that of Moser. Indeed, our result seems to give one of very few instances where a large scale computation of a numerical constant has an application.