ACW>In general, regardless of base, the period of 1/n divides phi(n). So perhaps it would be interesting to provide phi(n)/period. On Sat, May 25, 2013 at 6:32 PM, Hans Havermann <gladhobo@teksavvy.com>wrote: Bill Gosper: In[209]:= RealDigits[1/499999] Out[209]= A very large output was generated. Here is a sample of it: {{{2,0,0,0,0,0,4,0,0,0,0,0,8,0,0,0,0,1,6,0,0,0,0,3,2,0,0,0,0,6,4,0,0,0,1,2,8,0,0,0,2,5, <<26587>>, 5,1,5,6,2,5,0,3,1,2,5,0,0,6,2,5,0,0,1,2,5,0,0,0,2,5,0,0,0,0,5,0,0,0,0,1,0,0,0,0,0}},<<1>>} HH> Mathematica 9 just gives: A very large output was generated. Here is a sample of it: <<1>> RWG>I would have expected shorter periods. HH>I wanted to chart the period lengths for numbers of the form 49..9: Do[Print[Length[RealDigits[1/(10^n/2-1)][[1,1]]]],{n,2,16}] 42 498 357 641 26670 2499999 10248414 27737540 7001968 $Aborted This is better: Do[Print[MultiplicativeOrder[10,10^n/2-1]],{n,2,32}] 42 498 357 [...] -------------- FI[n_Integer] := Times @@ (HoldForm @@ {#[[1]]}^#[[2]] &) /@ FactorInteger[n] n 10^n/2-1 phi/period period In[230]:= TableForm[Table[Join[{n},FI/@{10^n/2-1,"\t",EulerPhi[10^n/2-1]/#,"\t",#}&@MultiplicativeOrder[10,10^n/2-1]/.FI->(#&)],{n,2,32}]] Out[230]//TableForm= 2 7^2 1 2 3 7 3 499 1 2 3 83 4 4999 2 7 3 7 17 5 49999 2 3 13 641 6 31 127^2 2 3^2 2 3 5 7 127 7 4999999 2 3 191 4363 8 7 23 310559 2^2 2 3 11 31 5009 9 691 723589 2 3^2 2^2 5 17 23 3547 10 17 14033 20959 2^5 3 7 2^4 499 877 11 29 1724137931 2 2^2 5 7 139 1240387 12 2969 168406871 2^4 5 7 53 3499 4813 13 491 10183299389 2 2^2 5 7^2 43 3041 19469 14 7 7142857142857 2 3 2^3 3 19 191 4363 18797 15 499999999999999 2 3 307 271444082519 16 17737 281896600327 2 3 739 2^3 3 11 739 5779649 17 19 2631578947368421 2 3 2^2 3^2 5 7 23 739 1187 310559 18 223 208513 10753058401 2^6 3^2 5 37 2^7 3^2 5 7 37 181 17299 19 1871 3061 873037477229 2^3 3 5 17 2^2 3 5 7^2 11 17 4454272843 20 7 782209 9131647862473 2^5 3^3 7 2^6 3 17^2 97 947 1390241 21 31 8999 1792313841322871 2^3 5 3 5 11 409 1789 57503 1742261 22 1049 2137 2230437170146223 2^4 2^3 3 89 131 1601 501187 1389853 23 1699 2069 3121 1382819 3295771 2^7 3^2 5 2^2 3 5 11 13 47 283 109859 691409 24 41717706103 11985318626233 2 3 2^3 3^3 7 587 1559 1316099 45760861 25 59 88369891 958989105371471 2^2 3 5 2 5 7 29 420809 95898910537147 26 7 17 420168067226890756302521 2^4 2^4 3 5 43 244283760015634160641 27 229 367021 409453351 14529115561 2^6 3^3 5^2 2^2 3^2 5^2 19 61 73 613 2039 121075963 28 4728835511159 1057342761912761 2^2 2^2 5 11 17 19 47 73 653 1109 3323 2130536717 29 71 76863049 9162079325173218881 2^6 3 5 2^4 5 7 47 4973 5741 12967 68141 77339 30 23 47 457 3691153 274199124029771599 2^6 3^2 2^4 3^2 7^2 11 19 23 25633 16760311 55646347 31 761 1295159 1798241 44167979 63871459 2^7 5 7 487 2^4 3 5 19 71 137 331 11239 21419 647579 32 7 7142857142857142857142857142857 2 3 2^3 3 19 23 389 739 1187 128509 310559 50010001 --rwg