Anomalous Cancellation Shalosh B. EKHAD and Doron ZEILBERGER

\frac{19̸5}{39̸0} = \frac{15}{30}

\frac{6̸4}{16̸0} = \frac{4}{10} \frac{9̸5}{19̸0} = \frac{5}{10} \frac{9̸6}{19̸2} = \frac{6}{12} \frac{9̸7}{19̸4} = \frac{7}{14} \frac{29̸1}{19̸4} = \frac{21}{14} \frac{9̸8}{19̸6} = \frac{8}{16} \frac{29̸4}{19̸6} = \frac{24}{16} \frac{49̸0}{19̸6} = \frac{40}{16} \frac{122̸1}{22̸2} = \frac{121}{22} \frac{134̸2}{24̸4} = \frac{132}{24} \frac{6̸5}{26̸0} = \frac{5}{20} \frac{66̸5}{26̸6} = \frac{65}{26} \frac{146̸3}{26̸6} = \frac{143}{26} \frac{226̸1}{26̸6} = \frac{221}{26} \frac{158̸4}{28̸8} = \frac{154}{28} \frac{9̸7}{29̸1} = \frac{7}{21} \frac{19̸4}{29̸1} = \frac{14}{21} \frac{9̸8}{29̸4} = \frac{8}{24} \frac{19̸6}{29̸4} = \frac{16}{24} \frac{39̸2}{29̸4} = \frac{32}{24} \frac{49̸0}{29̸4} = \frac{40}{24} \frac{83̸0}{33̸2} = \frac{80}{32} \frac{9̸8}{39̸2} = \frac{8}{32}

\frac{19̸6}{39̸2} = \frac{16}{32}

\frac{29̸4}{39̸2} = \frac{24}{32}

\frac{49̸0}{39̸2} = \frac{40}{32}

\frac{19̸7}{39̸4} = \frac{17}{34}

\frac{59̸1}{39̸4} = \frac{51}{34}

\frac{19̸9}{39̸8} = \frac{19}{38}

\frac{59̸7}{39̸8} = \frac{57}{38}

\frac{99̸5}{39̸8} = \frac{95}{38}

\frac{139̸3}{39̸8} = \frac{133}{38}

\frac{179̸1}{39̸8} = \frac{171}{38}

\frac{232̸1}{42̸2} = \frac{231}{42}

\frac{144̸3}{44̸4} = \frac{143}{44}

\frac{244̸2}{44̸4} = \frac{242}{44}

\frac{344̸1}{44̸4} = \frac{341}{44}

\frac{116̸0}{46̸4} = \frac{110}{44}

\frac{116̸5}{46̸6} = \frac{115}{46}

\frac{256̸3}{46̸6} = \frac{253}{46}

\frac{396̸1}{46̸6} = \frac{391}{46}

\frac{158̸6}{48̸8} = \frac{156}{48}

\frac{268̸4}{48̸8} = \frac{264}{48}

\frac{378̸2}{48̸8} = \frac{372}{48} \frac{19̸6}{49̸0} = \frac{16}{40}

\frac{29̸4}{49̸0} = \frac{24}{40}

\frac{39̸2}{49̸0} = \frac{32}{40}

\frac{13̸3}{53̸2} = \frac{13}{52}

\frac{93̸1}{53̸2} = \frac{91}{52}

\frac{133̸0}{53̸2} = \frac{130}{52}

\frac{155̸4}{55̸5} = \frac{154}{55}

\frac{255̸3}{55̸5} = \frac{253}{55}

\frac{355̸2}{55̸5} = \frac{352}{55}

\frac{455̸1}{55̸5} = \frac{451}{55}

\frac{187̸2}{57̸6} = \frac{182}{56}

\frac{29̸5}{59̸0} = \frac{25}{50}

\frac{19̸7}{59̸1} = \frac{17}{51}

\frac{39̸4}{59̸1} = \frac{34}{51}

\frac{29̸6}{59̸2} = \frac{26}{52}

\frac{29̸8}{59̸6} = \frac{28}{56}

\frac{89̸4}{59̸6} = \frac{84}{56}

\frac{149̸0}{59̸6} = \frac{140}{56}

\frac{19̸9}{59̸7} = \frac{19}{57}

\frac{39̸8}{59̸7} = \frac{38}{57}

\frac{79̸6}{59̸7} = \frac{76}{57}

\frac{99̸5}{59̸7} = \frac{95}{57} \frac{159̸2}{59̸7} = \frac{152}{57}

\frac{199̸0}{59̸7} = \frac{190}{57}

\frac{29̸9}{59̸8} = \frac{29}{58}

\frac{89̸7}{59̸8} = \frac{87}{58}

\frac{149̸5}{59̸8} = \frac{145}{58}

\frac{209̸3}{59̸8} = \frac{203}{58}

\frac{269̸1}{59̸8} = \frac{261}{58}

\frac{342̸1}{62̸2} = \frac{341}{62}

\frac{354̸2}{64̸4} = \frac{352}{64}

\frac{16̸6}{66̸4} = \frac{16}{64}

\frac{116̸2}{66̸4} = \frac{112}{64}

\frac{166̸0}{66̸4} = \frac{160}{64}

\frac{26̸6}{66̸5} = \frac{26}{65}

\frac{106̸4}{66̸5} = \frac{104}{65}

\frac{146̸3}{66̸5} = \frac{143}{65}

\frac{186̸2}{66̸5} = \frac{182}{65}

\frac{226̸1}{66̸5} = \frac{221}{65}

\frac{166̸5}{66̸6} = \frac{165}{66}

\frac{366̸3}{66̸6} = \frac{363}{66}

\frac{566̸1}{66̸6} = \frac{561}{66}

\frac{378̸4}{68̸8} = \frac{374}{68}

\frac{183̸0}{73̸2} = \frac{180}{72} \frac{177̸6}{77̸7} = \frac{176}{77}

\frac{277̸5}{77̸7} = \frac{275}{77}

\frac{377̸4}{77̸7} = \frac{374}{77}

\frac{477̸3}{77̸7} = \frac{473}{77}

\frac{577̸2}{77̸7} = \frac{572}{77}

\frac{677̸1}{77̸7} = \frac{671}{77}

\frac{39̸5}{79̸0} = \frac{35}{70}

\frac{39̸7}{79̸4} = \frac{37}{74}

\frac{119̸1}{79̸4} = \frac{111}{74}

\frac{19̸9}{79̸6} = \frac{19}{76}

\frac{39̸8}{79̸6} = \frac{38}{76}

\frac{59̸7}{79̸6} = \frac{57}{76}

\frac{99̸5}{79̸6} = \frac{95}{76}

\frac{119̸4}{79̸6} = \frac{114}{76}

\frac{139̸3}{79̸6} = \frac{133}{76}

\frac{179̸1}{79̸6} = \frac{171}{76}

\frac{199̸0}{79̸6} = \frac{190}{76}

\frac{39̸9}{79̸8} = \frac{39}{78}

\frac{119̸7}{79̸8} = \frac{117}{78}

\frac{199̸5}{79̸8} = \frac{195}{78}

\frac{279̸3}{79̸8} = \frac{273}{78}

\frac{359̸1}{79̸8} = \frac{351}{78} \frac{33̸2}{83̸0} = \frac{32}{80}

\frac{274̸3}{84̸4} = \frac{273}{84}

\frac{464̸2}{84̸4} = \frac{462}{84}

\frac{654̸1}{84̸4} = \frac{651}{84}

\frac{195̸2}{85̸4} = \frac{192}{84}

\frac{305̸0}{85̸4} = \frac{300}{84}

\frac{216̸0}{86̸4} = \frac{210}{84}

\frac{216̸5}{86̸6} = \frac{215}{86}

\frac{476̸3}{86̸6} = \frac{473}{86}

\frac{736̸1}{86̸6} = \frac{731}{86}

\frac{188̸7}{88̸8} = \frac{187}{88}

\frac{288̸6}{88̸8} = \frac{286}{88}

\frac{388̸5}{88̸8} = \frac{385}{88}

\frac{488̸4}{88̸8} = \frac{484}{88}

\frac{588̸3}{88̸8} = \frac{583}{88}

\frac{688̸2}{88̸8} = \frac{682}{88}

\frac{788̸1}{88̸8} = \frac{781}{88}

\frac{29̸8}{89̸4} = \frac{28}{84}

\frac{59̸6}{89̸4} = \frac{56}{84}

\frac{119̸2}{89̸4} = \frac{112}{84}

\frac{149̸0}{89̸4} = \frac{140}{84}

\frac{29̸9}{89̸7} = \frac{29}{87} \frac{119̸6}{89̸7} = \frac{116}{87}

\frac{149̸5}{89̸7} = \frac{145}{87}

\frac{209̸3}{89̸7} = \frac{203}{87}

\frac{239̸2}{89̸7} = \frac{232}{87}

\frac{299̸0}{89̸7} = \frac{290}{87}

\frac{13̸3}{93̸1} = \frac{13}{91}

\frac{53̸2}{93̸1} = \frac{52}{91}

\frac{133̸0}{93̸1} = \frac{130}{91}

\frac{23̸3}{93̸2} = \frac{23}{92}

\frac{163̸1}{93̸2} = \frac{161}{92}

\frac{233̸0}{93̸2} = \frac{230}{92}

\frac{207̸4}{97̸6} = \frac{204}{96}

\frac{317̸2}{97̸6} = \frac{312}{96}

\frac{427̸0}{97̸6} = \frac{420}{96}

\frac{49̸6}{99̸2} = \frac{46}{92}

\frac{49̸7}{99̸4} = \frac{47}{94}

\frac{149̸1}{99̸4} = \frac{141}{94}

\frac{19̸9}{99̸5} = \frac{19}{95}

\frac{39̸8}{99̸5} = \frac{38}{95}

\frac{59̸7}{99̸5} = \frac{57}{95}

\frac{79̸6}{99̸5} = \frac{76}{95}

\frac{119̸4}{99̸5} = \frac{114}{95} \frac{139̸3}{99̸5} = \frac{133}{95} \frac{159̸2}{99̸5} = \frac{152}{95} \frac{179̸1}{99̸5} = \frac{171}{95} \frac{49̸8}{99̸6} = \frac{48}{96} \frac{149̸4}{99̸6} = \frac{144}{96} \frac{249̸0}{99̸6} = \frac{240}{96} \frac{49̸9}{99̸8} = \frac{49}{98} \frac{149̸7}{99̸8} = \frac{147}{98} \frac{249̸5}{99̸8} = \frac{245}{98} \frac{349̸3}{99̸8} = \frac{343}{98} \frac{449̸1}{99̸8} = \frac{441}{98}

References

Shalosh B. Ekhad, c/o D. Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Email: ShaloshBEkhad at gmail dot com .

Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Email: DoronZeil at gmail dot com .