The procedure I suggested for approximating real x by rational a/b so that sqrt(a^2+b^2)=integer, was one of many possible and I do not claim my procedure is "best." It is merely good enough to produce an infinite set of approximations a/b which obey the constraint and are arbitrarily good. 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? For the non-pythagorean rational approximation problem P=2 is best possible and achieved by the usual continued fraction algorithm with the hardest-to-approximate number being the Golden number. Now that the Carsten Elsner: Fibonacci Quarterly (May 2003) 98-104 paper has been pointed out, it appears to have pretty much the same ideas that I used, and he demonstrates in coro 1.1 that P>=1 is valid for almost all irrational x, and also demonstrates in coro 1.2 that P=1 is best possible i.e that there exist quadratic-irrational x such that always error>PositiveConstant*|b|^(-1). I think the procedure I had described achieved P=1 in which case it actually was best possible in that weak sense (although you should check this numerically please).