What do High-Water Marks in Continued Fractions mean? While reading through several articles concerned with mathematical constants, I kept on finding things like this: > The continued fraction for $\mu$(Soldner's Constant) is given by $\left[1, 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, ...\right]$. > > The **high-water marks** are 1, 2, 4, 47, 99, 294, 527, 616, 1152, ... , which occur at positions 1, 2, 3, 12, 70, 126, 202, 585, 1592, ... . (copied from here) I didn't find a definition of **high-water marks** in the web, so I assume that it's a listing of increasing largest integers, while going through the continued fraction expansion. Is this correct and is there special meaing behind them?

Your definition seems correct to me -- at least, it agrees with the data you've provided and with my intuition, as a native speaker of English, of how this phrase is used. I don't know of a specific significance of these high-water marks, but it's well-known that cutting off a continued fraction expansion just before a particularly large coefficient gives a good rational approximation to the number being expanded. For example the sequence of convergents of π begins 3, 7, 15, 1, 292; the continued fraction [3, 7, 15, 1] is the well-known, surprisingly good approximation 355/113.