Re: [math-fun] pythagorean rationals & approximations by them

It is clear this procedure will cause approximation error=|x - a/b| which will be upper bounded by some negative power of |b|, say error<PositiveConstant*|b|^(-P), for an infinite set of (a,b).

A natural question would be: what is the greatest P that can be achieved?

--also note P will depend upon the number x being approximated. Although P=1 is greatest possible in general, it is quite possible that many particular x, such as x=pi, might enjoy some greater P (like P=2.54 or something) i.e. are far better approximable by pythagorean rationals. QUESTION: find such an x, or prove none exist.