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 Thébault, in volume $19$ of Scripta Mathematica.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $729$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $729$