## 2. Property of pattern of repeating decimal

### A consecution of "9"

For example, cutting 142857, repeating sequence of 1/7, at the center and adding 142 and 857 then 999 appears.
It always occurs when the length of sequence is even.

``` 7 : 142 + 857 = 9
11 : 0 + 9 = 9
13 : 076 + 923 = 999
17 : 05882352 + 94117647 = 99999999
19 : 052631578 + 947368421 = 999999999
23 : 04347826086 + 95652173913 = 99999999999
29 : 03448275862068 + 96551724137931 = 99999999999999
47 : 02127659574468085106382 + 97872340425531914893617 = 99999999999999999999999
59 : 01694915254237288135593220338 + 98305084745762711864406779661 = 99999999999999999999999999999
61 : 016393442622950819672131147540 + 983606557377049180327868852459 = 999999999999999999999999999999
73 : 0136 + 9863 = 9999
89 : 0112359550561797752808 + 9887640449438202247191 = 9999999999999999999999
97 : 010309278350515463917525773195876288659793814432 + 989690721649484536082474226804123711340206185567 = 999999999999999999999999999999999999999999999999
```

When the length is composite number, it has the same property.

``` 7 : 14 + 28 + 57 = 99
13 : 07 + 69 + 23 = 99
19 : 052631 + 578947 + 368421 = 999999
31 : 03225 + 80645 + 16129 = 99999  ,  032 + 258 + 064 + 516 + 129 = 999
37 : 0 + 2 + 7 = 9
43 : 0232558 + 1395348 + 8372093 = 9999999
61 : 01639344262295081967 + 21311475409836065573 + 77049180327868852459 = 99999999999999999999
67 : 01492537313 + 43283582089 + 55223880597 = 99999999999
97 : 01030927835051546391752577319587 + 62886597938144329896907216494845 + 36082474226804123711340206185567 = 99999999999999999999999999999999
```

When the length of sequence is prime, it does not have this property
(because we cannot divide by any number except p and 1).

Example : p=41 (length=5), p=53 (length=13), p=79 (length=13), p=83 (length=41), ...

In case the length is composite number, it sometimes does not have this property.

Example : p=71 (length 35)

5 sections : 0140845 + 0704225 + 3521126 + 7605633 + 8028169 = 19999998
7 sections : 01408 + 45070 + 42253 + 52112 + 67605 + 63380 + 28169 = 299997

### When the length of repeating sequence is p-1 ...

• cyclic patterns of its repeating sequence appear by multiplying 2, 3, ... , p-1
• a consecutive "9" appears by multiplying by p

Example : multiplying 142857 and 2, 3, 4, 5, 6, then cyclic patterns of 142857 appear.

142857 * 2 = 285714
142857 * 3 = 428571
142857 * 4 = 571428
142857 * 5 = 714285
142857 * 6 = 857142

multiplying by 7 (p itself), consecutive "9" appears.

142857 * 7 = 999999

Under 100, when p=7, 17, 19, 23, 29, 47, 59, 61, 97,
the length of repeating sequence is p-1, and these number have this property.

Example : In case 1/17 : 0588235294117647

588235294117647 *  2 = 1176470588235294
588235294117647 *  3 = 1764705882352941
588235294117647 *  4 = 2352941176470588
588235294117647 *  5 = 2941176470588235
588235294117647 *  6 = 3529411764705882
588235294117647 *  7 = 4117647058823529
588235294117647 *  8 = 4705882352941176
588235294117647 *  9 = 5294117647058823
588235294117647 * 10 = 5882352941176470
588235294117647 * 11 = 6470588235294117
588235294117647 * 12 = 7058823529411764
588235294117647 * 13 = 7647058823529411
588235294117647 * 14 = 8235294117647058
588235294117647 * 15 = 8823529411764705
588235294117647 * 16 = 9411764705882352
and
588235294117647 * 17 = 9999999999999999

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E-mail : kc2h-msm@asahi-net.or.jp
Hisanori Mishima