Lois régissant les cycles des digits des quotients dans les divisions.

[diasoluyalu]

Bonjour chers amis @FS,

En m'amusant à analyser les parties fractionnaires cycliques de certains quotients, j'ai constaté que dans tous les cas le report est respecté, mais malgré le respect du report, le cycle bien que décalé, reste intact pour tous les numérateurs, même les multiples d'autres numérateurs.

Par exemple :
3/11 = 0.2727272727272727
6/11 = 0.5454545454545454
Dans ce cas, 6 est le double de 3, le report est respecté (7×2 = 14, 2×2=4 [+1=5]),mais le cycle n'est pas le même.

Considérons seulement les cycles de plus de 3 digits :

Pour la division par 7 (et ses multiples entiers)
x/7
0/7 = 0
1/7 = 0.14285714285714285
2/7 = 0.2857142857142857
3/7 = 0.42857142857142855
4/7 = 0.5714285714285714
5/7 = 0.7142857142857143
6/7 = 0.8571428571428571
7/7 = 1
8/7 = 1.1428571428571428
Dans ce cas de division par 7, le report est respecté quand on multiplie la partie cyclique « 142857 » de la partie fractionnaire par le facteur approprié, mais le cycle reste le même tout en étant simplement décallé.

Pour la division par 13 (et ses multiples entiers), il y a deux cycles le « 769230 » et le « 153846 ».
Le cycle « 769230 » revient pour les numérateurs
1, 3, 4, 9, 10, 12
14, 16, 17, 22, 23, 25
...
(des multiplies espacés de 13)

Le cycle « 153846 » revient pour les numérateurs
2, 5, 6, 7, 8, 11,
15, 18, 19, 20, 21, 24,
...

Pour la division par 14 (et ses multiples entiers), le cycle « 714285 » revient avec tous les numérateurs sans bien entendu les multiples de 14.

Pour la division par 17 (toujours avec leurs multiples entiers), le cycle « 5882352941176470 » revient avec tous les numérateurs sans bien entendu les multiples de 17.

Pour la division par 19, le cycle est « 5263157894736842 ».

Pour la division par 21, les cycles sont « 476190 », « 9523809 », « 142857 »

Pour la division par 23, le cycle « 434782608695652121739130 »

Pour la division par 29, « 3448275862068965517241379310 »

Pour la division par 31, « 322580645161290 », « 967741935483870 »

...

Pour la division par 67, « 149253731343283582089552238805 970 »

...etceteri, etcetera, etcetiroble, patati, patata, ainsi de suite...

**************************
* Quelqu'un pourrait-il m'expliquer
*
* 1. Le pourquoi de cela (le report dans la multiplication, mais qui ne perturbe pas le cycle) ?
*
* 2. Ce qui détermine la longueur du cycle, si possible avec un algorithme.
**************************

Merci.
-----

[diasoluyalu]

Modèle de code qui génère ces quotients.

<script>
tmp=""
for(m=0;m<=100;m++){
tmp+="x/"+m+"<br>"
for(k=0;k<=25;k++){
tmp+=k+"/"+m+" = "+k/m+"<br>"
}
tmp+="===============<br>"
}
document.write(tmp)
</script>


**-*-*-*-*-*--*-*-*-*-**


Voici des quotients de divisions.
Dommage que JavaScript n'affiche que 17 chiffres significatifs.

x/3
0/3 = 0
1/3 = 0.3333333333333333
2/3 = 0.6666666666666666
3/3 = 1
4/3 = 1.3333333333333333
5/3 = 1.6666666666666667
6/3 = 2
7/3 = 2.3333333333333335
8/3 = 2.6666666666666665
9/3 = 3
10/3 = 3.3333333333333335
11/3 = 3.6666666666666665
12/3 = 4
13/3 = 4.333333333333333
14/3 = 4.666666666666667
15/3 = 5
16/3 = 5.333333333333333
17/3 = 5.666666666666667
18/3 = 6
19/3 = 6.333333333333333
20/3 = 6.666666666666667
21/3 = 7
22/3 = 7.333333333333333
23/3 = 7.666666666666667
24/3 = 8
25/3 = 8.333333333333334
===============
x/6
0/6 = 0
1/6 = 0.16666666666666666
2/6 = 0.3333333333333333
3/6 = 0.5
4/6 = 0.6666666666666666
5/6 = 0.8333333333333334
6/6 = 1
7/6 = 1.1666666666666667
8/6 = 1.3333333333333333
9/6 = 1.5
10/6 = 1.6666666666666667
11/6 = 1.8333333333333333
12/6 = 2
13/6 = 2.1666666666666665
14/6 = 2.3333333333333335
15/6 = 2.5
16/6 = 2.6666666666666665
17/6 = 2.8333333333333335
18/6 = 3
19/6 = 3.1666666666666665
20/6 = 3.3333333333333335
21/6 = 3.5
22/6 = 3.6666666666666665
23/6 = 3.8333333333333335
24/6 = 4
25/6 = 4.166666666666667
===============
x/7
0/7 = 0
1/7 = 0.14285714285714285
2/7 = 0.2857142857142857
3/7 = 0.42857142857142855
4/7 = 0.5714285714285714
5/7 = 0.7142857142857143
6/7 = 0.8571428571428571
7/7 = 1
8/7 = 1.1428571428571428
9/7 = 1.2857142857142858
10/7 = 1.4285714285714286
11/7 = 1.5714285714285714
12/7 = 1.7142857142857142
13/7 = 1.8571428571428572
14/7 = 2
15/7 = 2.142857142857143
16/7 = 2.2857142857142856
17/7 = 2.4285714285714284
18/7 = 2.5714285714285716
19/7 = 2.7142857142857144
20/7 = 2.857142857142857
21/7 = 3
22/7 = 3.142857142857143
23/7 = 3.2857142857142856
24/7 = 3.4285714285714284
25/7 = 3.5714285714285716
===============
x/13
0/13 = 0
1/13 = 0.07692307692307693
2/13 = 0.15384615384615385
3/13 = 0.23076923076923078
4/13 = 0.3076923076923077
5/13 = 0.38461538461538464
6/13 = 0.46153846153846156
7/13 = 0.5384615384615384
8/13 = 0.6153846153846154
9/13 = 0.6923076923076923
10/13 = 0.7692307692307693
11/13 = 0.8461538461538461
12/13 = 0.9230769230769231
13/13 = 1
14/13 = 1.0769230769230769
15/13 = 1.1538461538461537
16/13 = 1.2307692307692308
17/13 = 1.3076923076923077
18/13 = 1.3846153846153846
19/13 = 1.4615384615384615
20/13 = 1.5384615384615385
21/13 = 1.6153846153846154
22/13 = 1.6923076923076923
23/13 = 1.7692307692307692
24/13 = 1.8461538461538463
25/13 = 1.9230769230769231
===============
x/14
0/14 = 0
1/14 = 0.07142857142857142
2/14 = 0.14285714285714285
3/14 = 0.21428571428571427
4/14 = 0.2857142857142857
5/14 = 0.35714285714285715
6/14 = 0.42857142857142855
7/14 = 0.5
8/14 = 0.5714285714285714
9/14 = 0.6428571428571429
10/14 = 0.7142857142857143
11/14 = 0.7857142857142857
12/14 = 0.8571428571428571
13/14 = 0.9285714285714286
14/14 = 1
15/14 = 1.0714285714285714
16/14 = 1.1428571428571428
17/14 = 1.2142857142857142
18/14 = 1.2857142857142858
19/14 = 1.3571428571428572
20/14 = 1.4285714285714286
21/14 = 1.5
22/14 = 1.5714285714285714
23/14 = 1.6428571428571428
24/14 = 1.7142857142857142
25/14 = 1.7857142857142858
===============
x/17
0/17 = 0
1/17 = 0.058823529411764705
2/17 = 0.11764705882352941
3/17 = 0.17647058823529413
4/17 = 0.23529411764705882
5/17 = 0.29411764705882354
6/17 = 0.35294117647058826
7/17 = 0.4117647058823529
8/17 = 0.47058823529411764
9/17 = 0.5294117647058824
10/17 = 0.5882352941176471
11/17 = 0.6470588235294118
12/17 = 0.7058823529411765
13/17 = 0.7647058823529411
14/17 = 0.8235294117647058
15/17 = 0.8823529411764706
16/17 = 0.9411764705882353
17/17 = 1
18/17 = 1.0588235294117647
19/17 = 1.1176470588235294
20/17 = 1.1764705882352942
21/17 = 1.2352941176470589
22/17 = 1.2941176470588236
23/17 = 1.3529411764705883
24/17 = 1.411764705882353
25/17 = 1.4705882352941178
===============
x/19
0/19 = 0
1/19 = 0.05263157894736842
2/19 = 0.10526315789473684
3/19 = 0.15789473684210525
4/19 = 0.21052631578947367
5/19 = 0.2631578947368421
6/19 = 0.3157894736842105
7/19 = 0.3684210526315789
8/19 = 0.42105263157894735
9/19 = 0.47368421052631576
10/19 = 0.5263157894736842
11/19 = 0.5789473684210527
12/19 = 0.631578947368421
13/19 = 0.6842105263157895
14/19 = 0.7368421052631579
15/19 = 0.7894736842105263
16/19 = 0.8421052631578947
17/19 = 0.8947368421052632
18/19 = 0.9473684210526315
19/19 = 1
20/19 = 1.0526315789473684
21/19 = 1.105263157894737
22/19 = 1.1578947368421053
23/19 = 1.2105263157894737
24/19 = 1.263157894736842
25/19 = 1.3157894736842106
===============
x/21
0/21 = 0
1/21 = 0.047619047619047616
2/21 = 0.09523809523809523
3/21 = 0.14285714285714285
4/21 = 0.19047619047619047
5/21 = 0.23809523809523808
6/21 = 0.2857142857142857
7/21 = 0.3333333333333333
8/21 = 0.38095238095238093
9/21 = 0.42857142857142855
10/21 = 0.47619047619047616
11/21 = 0.5238095238095238
12/21 = 0.5714285714285714
13/21 = 0.6190476190476191
14/21 = 0.6666666666666666
15/21 = 0.7142857142857143
16/21 = 0.7619047619047619
17/21 = 0.8095238095238095
18/21 = 0.8571428571428571
19/21 = 0.9047619047619048
20/21 = 0.9523809523809523
21/21 = 1
22/21 = 1.0476190476190477
23/21 = 1.0952380952380953
24/21 = 1.1428571428571428
25/21 = 1.1904761904761905
===============
x/23
0/23 = 0
1/23 = 0.043478260869565216
2/23 = 0.08695652173913043
3/23 = 0.13043478260869565
4/23 = 0.17391304347826086
5/23 = 0.21739130434782608
6/23 = 0.2608695652173913
7/23 = 0.30434782608695654
8/23 = 0.34782608695652173
9/23 = 0.391304347826087
10/23 = 0.43478260869565216
11/23 = 0.4782608695652174
12/23 = 0.5217391304347826
13/23 = 0.5652173913043478
14/23 = 0.6086956521739131
15/23 = 0.6521739130434783
16/23 = 0.6956521739130435
17/23 = 0.7391304347826086
18/23 = 0.782608695652174
19/23 = 0.8260869565217391
20/23 = 0.8695652173913043
21/23 = 0.9130434782608695
22/23 = 0.9565217391304348
23/23 = 1
24/23 = 1.0434782608695652
25/23 = 1.0869565217391304
===============
x/26
0/26 = 0
1/26 = 0.038461538461538464
2/26 = 0.07692307692307693
3/26 = 0.11538461538461539
4/26 = 0.15384615384615385
5/26 = 0.19230769230769232
6/26 = 0.23076923076923078
7/26 = 0.2692307692307692
8/26 = 0.3076923076923077
9/26 = 0.34615384615384615
10/26 = 0.38461538461538464
11/26 = 0.4230769230769231
12/26 = 0.46153846153846156
13/26 = 0.5
14/26 = 0.5384615384615384
15/26 = 0.5769230769230769
16/26 = 0.6153846153846154
17/26 = 0.6538461538461539
18/26 = 0.6923076923076923
19/26 = 0.7307692307692307
20/26 = 0.7692307692307693
21/26 = 0.8076923076923077
22/26 = 0.8461538461538461
23/26 = 0.8846153846153846
24/26 = 0.9230769230769231
25/26 = 0.9615384615384616
===============
x/27
0/27 = 0
1/27 = 0.037037037037037035
2/27 = 0.07407407407407407
3/27 = 0.1111111111111111
4/27 = 0.14814814814814814
5/27 = 0.18518518518518517
6/27 = 0.2222222222222222
7/27 = 0.25925925925925924
8/27 = 0.2962962962962963
9/27 = 0.3333333333333333
10/27 = 0.37037037037037035
11/27 = 0.4074074074074074
12/27 = 0.4444444444444444
13/27 = 0.48148148148148145
14/27 = 0.5185185185185185
15/27 = 0.5555555555555556
16/27 = 0.5925925925925926
17/27 = 0.6296296296296297
18/27 = 0.6666666666666666
19/27 = 0.7037037037037037
20/27 = 0.7407407407407407
21/27 = 0.7777777777777778
22/27 = 0.8148148148148148
23/27 = 0.8518518518518519
24/27 = 0.8888888888888888
25/27 = 0.9259259259259259
===============
x/28
0/28 = 0
1/28 = 0.03571428571428571
2/28 = 0.07142857142857142
3/28 = 0.10714285714285714
4/28 = 0.14285714285714285
5/28 = 0.17857142857142858
6/28 = 0.21428571428571427
7/28 = 0.25
8/28 = 0.2857142857142857
9/28 = 0.32142857142857145
10/28 = 0.35714285714285715
11/28 = 0.39285714285714285
12/28 = 0.42857142857142855
13/28 = 0.4642857142857143
14/28 = 0.5
15/28 = 0.5357142857142857
16/28 = 0.5714285714285714
17/28 = 0.6071428571428571
18/28 = 0.6428571428571429
19/28 = 0.6785714285714286
20/28 = 0.7142857142857143
21/28 = 0.75
22/28 = 0.7857142857142857
23/28 = 0.8214285714285714
24/28 = 0.8571428571428571
25/28 = 0.8928571428571429
===============
x/29
0/29 = 0
1/29 = 0.034482758620689655
2/29 = 0.06896551724137931
3/29 = 0.10344827586206896
4/29 = 0.13793103448275862
5/29 = 0.1724137931034483
6/29 = 0.20689655172413793
7/29 = 0.2413793103448276
8/29 = 0.27586206896551724
9/29 = 0.3103448275862069
10/29 = 0.3448275862068966
11/29 = 0.3793103448275862
12/29 = 0.41379310344827586
13/29 = 0.4482758620689655
14/29 = 0.4827586206896552
15/29 = 0.5172413793103449
16/29 = 0.5517241379310345
17/29 = 0.5862068965517241
18/29 = 0.6206896551724138
19/29 = 0.6551724137931034
20/29 = 0.6896551724137931
21/29 = 0.7241379310344828
22/29 = 0.7586206896551724
23/29 = 0.7931034482758621
24/29 = 0.8275862068965517
25/29 = 0.8620689655172413
===============
x/31
0/31 = 0
1/31 = 0.03225806451612903
2/31 = 0.06451612903225806
3/31 = 0.0967741935483871
4/31 = 0.12903225806451613
5/31 = 0.16129032258064516
6/31 = 0.1935483870967742
7/31 = 0.22580645161290322
8/31 = 0.25806451612903225
9/31 = 0.2903225806451613
10/31 = 0.3225806451612903
11/31 = 0.3548387096774194
12/31 = 0.3870967741935484
13/31 = 0.41935483870967744
14/31 = 0.45161290322580644
15/31 = 0.4838709677419355
16/31 = 0.5161290322580645
17/31 = 0.5483870967741935
18/31 = 0.5806451612903226
19/31 = 0.6129032258064516
20/31 = 0.6451612903225806
21/31 = 0.6774193548387096
22/31 = 0.7096774193548387
23/31 = 0.7419354838709677
24/31 = 0.7741935483870968
25/31 = 0.8064516129032258
===============
x/53
0/53 = 0
1/53 = 0.018867924528301886
2/53 = 0.03773584905660377
3/53 = 0.05660377358490566
4/53 = 0.07547169811320754
5/53 = 0.09433962264150944
6/53 = 0.11320754716981132
7/53 = 0.1320754716981132
8/53 = 0.1509433962264151
9/53 = 0.16981132075471697
10/53 = 0.18867924528301888
11/53 = 0.20754716981132076
12/53 = 0.22641509433962265
13/53 = 0.24528301886792453
14/53 = 0.2641509433962264
15/53 = 0.2830188679245283
16/53 = 0.3018867924528302
17/53 = 0.32075471698113206
18/53 = 0.33962264150943394
19/53 = 0.3584905660377358
20/53 = 0.37735849056603776
21/53 = 0.39622641509433965
22/53 = 0.41509433962264153
23/53 = 0.4339622641509434
24/53 = 0.4528301886792453
25/53 = 0.4716981132075472
===============
x/67
0/67 = 0
1/67 = 0.014925373134328358
2/67 = 0.029850746268656716
3/67 = 0.04477611940298507
4/67 = 0.05970149253731343
5/67 = 0.07462686567164178
6/67 = 0.08955223880597014
7/67 = 0.1044776119402985
8/67 = 0.11940298507462686
9/67 = 0.13432835820895522
10/67 = 0.14925373134328357
11/67 = 0.16417910447761194
12/67 = 0.1791044776119403
13/67 = 0.19402985074626866
14/67 = 0.208955223880597
15/67 = 0.22388059701492538
16/67 = 0.23880597014925373
17/67 = 0.2537313432835821
18/67 = 0.26865671641791045
19/67 = 0.2835820895522388
20/67 = 0.29850746268656714
21/67 = 0.31343283582089554
22/67 = 0.3283582089552239
23/67 = 0.34328358208955223
24/67 = 0.3582089552238806
25/67 = 0.373134328358209

[stefjm]

Un début d'explications ici : https://fr.wikipedia.org/wiki/D%C3%A...a_p.C3.A9riode