On Sun, Jan 29, 2012 at 9:19 AM, Warren Smith <warren.wds@gmail.com> wrote:
The idea that we can identify numbers like 1.004009016025036... by use of "generating functions" is silly in comparison with the much better and simpler idea that you can find the optimum rational approximations to ANY decimal X by use of (Euclid's) continued fraction expansion of X: 1.004009016025036049064081100 = [1, 249, 2, 3, 1, 1, 14, 7, 4, 2, 4, 7, 7, 1, large] then 1 + 1/(249+1/(2+1/(3+1/(1+1/(1+1/(14+1/(7+1/(4+1/(2+1/(4+1/(7+1/(7+1)))))))))))); = 1001000000 / 997002999 = 1.004009016025036049064081100121144169196225256289324361...
But without the "silly" generating functions, do we have any explanation of why if we take the continued fraction expansion of 1.004009016025036049064081100, and truncate before the first large entry, we end up with a decimal that continues with 121144169196225256289324361... Surely this is not coincidence! Continued fractions give us an efficient way to find the smallest-denominatored fraction that starts with the first ten three-digit squares, but unless I'm missing something, they give no insight as to why this expansion should continue with the next 10 3-digit squares, and the generating function approach does. Andy Latto