Period of Reciprocal of 729 is 81

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Theorem

The decimal expansion of the reciprocal of $729$ has $\dfrac 1 9$ the maximum period, that is, $81$:

$\dfrac 1 {729} = 0 \cdotp \dot 00137 \, 17421 \, 12482 \, 85322 \, 35939 \, 64334 \, 70507 \, 54458 \, 16186 \, 55692 \, 72976 \, 68038 \, 40877 \, 91495 \, 19890 \, 26063 \, \dot 1$


The recurring part can be arranged in groups of $9$ digits each, revealing an interesting pattern:

\(\ds 001 \, 371 \, 742\) \(\) \(\ds \)
\(\ds 112 \, 482 \, 853\) \(\) \(\ds \)
\(\ds 223 \, 593 \, 964\) \(\) \(\ds \)
\(\ds 334 \, 705 \, 075\) \(\) \(\ds \)
\(\ds 445 \, 816 \, 186\) \(\) \(\ds \)
\(\ds 556 \, 927 \, 297\) \(\) \(\ds \)
\(\ds 668 \, 038 \, 408\) \(\) \(\ds \)
\(\ds 779 \, 149 \, 519\) \(\) \(\ds \)
\(\ds 890 \, 260 \, 631\) \(\) \(\ds \)

that is, each row (apart from the last) can be obtained from the previous one by adding $111 \, 111 \, 111$ to it.

This sequence is A021733 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Performing the calculation using long division:

    0.00137174211248285322359396433470507544581618655692729766803840877914951989026063100
   ---------------------------------------------------------------------------------------
729)1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000
      729    729   3645   2187   5103     5832   3645   5103     5832   3645     4374
      ---    ---   ----   ----   ----     ----   ----   ----     ----   ----     ----
      2710    910   2350   7030    3700    1180   5050   4870     5680   1450     2260
      2187    729   2187   6561    3645     729   4374   4374     5103    729     2187
      ----    ---   ----   ----    ----    ----   ----   ----     ----   ----     ----
       5230   1810   1630   4690     5500   4510   6760   4960     5770   7210      730
       5103   1458   1458   4374     5103   4374   6561   4374     5103   6561      729
       ----   ----   ----   ----     ----   ----   ----   ----     ----   ----      ---
        1270   3520   1720   3160     3970   1360   1990   5860     6670   6490       1000
         729   2916   1458   2916     3645    729   1458   5832     6561   5832        729
        ----   ----   ----   ----     ----   ----   ----   ----     ----   ----       ----
         5410   6040   2620   2440     3250   6310   5320    2800    1090   6580       ...
         5103   5832   2187   2187     2916   5832   5103    2187     729   6561
         ----   ----   ----   ----     ----   ----   ----    ----    ----   ----
          3070   2080   4330   2530     3340   4780   2170    6130    3610    1900
          2916   1458   3645   2187     2916   4374   1458    5832    2916    1458
          ----   ----   ----   ----     ----   ----   ----    ----    ----    ----
           1540   6220   6850   3430     4240   4060   7120    2980    6940    4420
           1458   5832   6561   2916     3645   3645   6561    2916    6561    4374
           ----   ----   ----   ----     ----   ----   ----    ----    ----    ----
             820   3880   2890   5140     5950   4150   5590     6400   3790     4600
             729   3645   2187   5103     5832   3645   5103     5832   3645     4374

$\blacksquare$


Historical Note

According to David Wells, in his Curious and Interesting Numbers of $1986$, this observation was reported on by Victor Th├ębault, in volume $19$ of Scripta Mathematica.


Sources