Superficial observations. How much does rounding (thereby disabling terms =1, and enabling negative terms) shorten the period? In[281]:= rtcfperiod /@ Range[69] - rndcfperiod /@ Range[69] Out[281]= {{0}, 0, 0, {1}, 0, 0, 2, 0, {2}, 0, 0, 0, 2, 2, 0, {3}, 0, \ 0, 2, 0, 2, 2, 2, 0, {4}, 0, 0, 0, -3, 0, 2, 2, 2, 2, 0, {5}, 0, 0, \ 0, 0, 0, 0, 4, 4, 2, 4, 2, 0, {6}, 0, 0, 2, -3, 2, 0, 0, 2, -1, 2, 2, \ 4, 2, 0, {7}, 0, 0, 2, 0, 2} It actually lengthened √29 (and √53) by three! and √241 by seven: In[282]:= rtcfperiod /@ Range[239, 288] - rndcfperiod /@ Range[239, 288] Out[282]= {2, 0, -7, 4, 2, 12, 4, 4, -2, 2, 6, 2, 4, 2, 8, 2, 0, \ {15}, 0, 0, 2, 0, 4, 4, 4, 0, 2, 0, 2, 6, 0, 0, 8, 0, 2, 2, 2, 4, 6, \ 2, -2, 0, 4, 4, 8, 6, 2, 2, 2, 0} The biggest helps seem to outweigh the biggest hurts. The curlybraces mark the terminating (square) cases. In[275]:= cfrnt[29, 9] Out[275]= {{{0, 29}, {1, 0}}, {{5, 1}, {4, -5}}, {{7, 4}, {-5, -7}}, {{3, -5}, {-4, -3}}, {{5, -4}, {-1, -5}}, {{5, -1}, {-4, -5}}, {{7, -4}, {5, -7}}, {{3, 5}, {4, -3}}, {{5, 4}, {1, -5}}, {{5, 1}, {4, -5}}} The actual √29 roundcf, In[276]:= rndtrm /@ % Out[276]= {5, 3, -2, -2, -10, -3, 2, 2, 10, 3} vs the vanilla one, In[278]:= ContinuedFraction[Sqrt[29], Length[%%]] Out[278]= {5, 2, 1, 1, 2, 10, 2, 1, 1, 2} In[279]:= FromContinuedFraction /@ {%276, %278} Out[279]= {315156/58523, 9801/1820} of course converges faster, term for term: In[280]:= % - Sqrt[29.] Out[280]= {-1.35546*10^-10, 2.80303*10^-8} Even faster, period for period. --rwg