[DanA wrote] PS Let me add some afterthoughts: ----- Question: --------- Are there algorithms relating the cfe's of x and y with the cfe of simple functions of x and y like x + y x - y x * y x / y x ^ y —Dan Dan, it's grim: In[257]:= Coth /@ {1/2, 1, 2} Out[257]= {Coth[1/2], Coth[1], Coth[2]} These coths are related by the double angle formula (1+c^2)/2/c. But look at the CFs: In[258]:= ContinuedFraction[#, 5] & /@ % Out[258]= {{2, 6, 10, 14, 18}, {1, 3, 5, 7, 9}, {1, 26, 1, 3, 1}} Coth 2 evidently = 1/2,3/2,5/2,..., but normalizing pseudorandomizes: In[301]:= ContinuedFraction[FromContinuedFraction[{1, 3, 5, 7, 9}/2]] Out[301]= {1, 26, 1, 4, 2, 2, 1, 2} Unlike the linear conversions in my old http://www.tweedledum.com/rwg/cfup.htm, even the spigot conversion of x={2, 6, 10, 14, 18,...} to (1+x^2)/2/x = {1, 3, 5, 7, 9,..} is non finite state: Initialize V(x) to the symbolic expression (1+x^2)/2/x . Input terms via x → 2+1/x, 6+1/x, ... . Output t =⎣V(1)⎦ when it equals ⎣V(∞)⎦. V → 1/(V - t). (This is easy. Try it.) Input and output will strictly alternate until the input of 26 and the output of 13 leaves V = (2 (1654632 + 86200843 x + 1122694538 x^2))/ (229783 + 11709446 x + 149100967 x^2) (!) which can further output 15(!) This was unavoidable. The larger terms of the input contributed information faster than the smaller output terms were removing it. For a conversion to be finite state, the input and output information densities must match exactly. There might be (necessarily non-unique) number representations less efficient than decimal or cf that might reveal number-theoretic secrets. —rwg