Suppose we have a mysterious quantity foo. We can quickly detect its Hurwitz signature: In[44]:= hursig[ContinuedFraction[foo, 69]] Out[44]= {1, 1, 1, 4 k, 3, 2 + 4 k} This is not a single mover, so we can't use fromcf. Propose successively larger homographic determinants 2, 3, ... In[45]:= hursigs[foo, 69, 2] During evaluation of In[45]:= 0.151464 Out[45]= {3, 2 -2 18 {1, 1, 1, 4 + 6 k, 1, 4, 1, 5 + 6 k, 1, 1, 1, 29 + 24 k, } -2 3 3, 8 + 6 k, 6, 9 + 6 k, 3, 41 + 24 k} 1 0 18 {1, 1, 1, 6 k, 1, 4, 1, 1 + 6 k, 1, 1, 1, 13 + 24 k, 3, -1 2 4 + 6 k, 6, 5 + 6 k, 3, 25 + 24 k} 1 -1 18 {1, 1, 1, 2 + 6 k, 1, 4, 1, 3 + 6 k, 1, 1, 1, 21 + 24 k, -2 4 3, 6 + 6 k, 6, 7 + 6 k, 3, 33 + 24 k} No single movers--fail. In[46]:= hursigs[foo, 69, 3] During evaluation of In[46]:= 0.27556 Out[46]= {6, 3 -3 22 {1, 1, 1 + 4 k, 9, 2 + 4 k, 4, 1, 1, 2 + 4 k, 2, 3, 1, -3 4 2 + 4 k, 1, 7, 1, 3 + 4 k, 2, 3, 1, 4 + 4 k, 4} 1 3 16 {1, 1, 4, 9 + 3 k, 2 + k, 3, 1 - k, 9 + 2 k, 1 - 3 k, 1, 1 6 4 + 3 k, 10 + 3 k, 1, 3, 2, 11 + 3 k} 1 1 2 {1, 1 + 6 k} -1 2 1 -1 20 {1, 1, 4, 4 k, 9, 4 k, 1, 7, 1, 4 k, 1, 3, 2, 1 + 4 k, 1, -3 6 7, 1, 2 + 4 k, 9, 3 + 4 k} 1 -3 16 {1, 3, 2, 8 + 3 k, 1 + 3 k, 1, 4, 9 + 3 k, 2 + k, 3, 3 -6 1 - k, 9 + 2 k, 1 - 3 k, 1, 4 + 3 k, 10 + 3 k} 1 -3 12 {1, 1 - 2 k, 4 + 2 k, 10 + 2 k, 1, 3 + 2 k, 2 - 2 k, 2 -3 11 + 2 k, 4 + 2 k, 1 - 2 k, 1, 11 + 2 k} } Bingo in the third row. In[48]:= %46[[2, 1, 3]] Out[48]= {1 1 , 2, {1, 1 + 6 k}} -1 2 In[50]:= Simplify[mat2fn[Inverse[%%[[1, 1]]]]@fromcf[%%[[3]], k]] Out[50]= (1 + Tan[1/3])/(2 - Tan[1/3]) In[51]:= ContinuedFraction[#, 22] & /@ {%, foo} Out[51]= {{0, 1, 4, 2, 1, 1, 1, 4, 3, 6, 1, 1, 1, 8, 3, 10, 1, 1, 1, 12, 3, 14}, {1, 4, 2, 1, 1, 1, 4, 3, 6, 1, 1, 1, 8, 3, 10, 1, 1, 1, 12, 3, 14, 1}} So foo == (2*Cos[1/3] - Sin[1/3])/(Cos[1/3] + Sin[1/3]) I've yet to see a "naturally occurring" Hurwitz number that this scheme won't catch, i.e., reduce to a single mover. But if need be, the hursigs function could be generalized to handle Hurwitzes with quadratic progressors, or even higher. I have no idea how to generalize fromcf. --rwg