Contents

Introduction

On other pages we've seen how the 's' values, the potential numerators found while searching for loops in 3x+1, are actually loop seeds in other systems (like 3x+5). Mostly though, these 's' values are seeds in systems with very large 'k' values. Loops in systems with small 'k' values usually derive from systems where 's' and 'K' have a common factor (f), so k = K∕f and the seed is s∕f.

I've ignored systems where k is a multiple of 3 as they aren't covered by the equations considered on this site (as 2m−3n can never be a multiple of 3). 

I ran a search for cycles in systems with k values up to 200 (ignoring multiples of 2 and 3) and seed values up to 100,000,000. The largest seed I actually found was 35,275 in 3x+145 and the largest cycle had 80 members, in 3x+143.

It looks like most if not all cycles are in the domain where s/k is relatively small. 's' is the smallest member of the loop, other members can be relatively large.

Loops in small k values

This section shows some of the loops discovered in systems with k values up to 19. The largest loop is 18 in 3x+17, the largest seed is 347 in 3x+5, and the largest s∕k ratio is 69.4 in 3x+5.

Loops in 3x+1

SeedSizeSeed/KLoop
111.0[1, 1]

Loops in 3x+5

SeedSizeSeed/KLoop
110.2[1, 1]
511.0[5, 5]
1933.8[19, 31, 49, 19]
2334.6[23, 37, 29, 23]
1871737.4[187, 283, 427, 643, 967, 1453, 1091, 1639, 2461, 1847, 2773, 2081, 781, 587, 883, 1327, 1993, 187]
3471769.4[347, 523, 787, 1183, 1777, 667, 1003, 1507, 2263, 3397, 2549, 1913, 359, 541, 407, 613, 461, 347]

Loops in 3x+7

SeedSizeSeed/KLoop
520.7[5, 11, 5]
711.0[7, 7]

Loops in 3x+11

SeedSizeSeed/KLoop
120.1[1, 7, 1]
1111.0[11, 11]
1381.2[13, 25, 43, 35, 29, 49, 79, 31, 13]

Loops in 3x+13

SeedSizeSeed/KLoop
110.1[1, 1]
1311.0[13, 13]
1311510.1[131, 203, 311, 473, 179, 275, 419, 635, 959, 1445, 1087, 1637, 1231, 1853, 1393, 131]
211516.2[211, 323, 491, 743, 1121, 211]
227517.5[227, 347, 527, 797, 601, 227]
251519.3[251, 383, 581, 439, 665, 251]
259519.9[259, 395, 599, 905, 341, 259]
283521.8[283, 431, 653, 493, 373, 283]
287522.1[287, 437, 331, 503, 761, 287]
319524.5[319, 485, 367, 557, 421, 319]

Loops in 3x+17

SeedSizeSeed/KLoop
120.1[1, 5, 1]
1711.0[17, 17]
23181.4[23, 43, 73, 59, 97, 77, 31, 55, 91, 145, 113, 89, 71, 115, 181, 35, 61, 25, 23]

Loops in 3x+19

SeedSizeSeed/KLoop
550.3[5, 17, 35, 31, 7, 5]
1911.0[19, 19]

Larger 'k' values

This table summarises the results up to k=199.

KSeedSizeSeed/K
1111.0
5110.2
5511.0
51933.8
52334.6
51871737.4
53471769.4
7520.7
7711.0
11120.1
111111.0
111381.2
13110.1
131311.0
131311510.1
13211516.2
13227517.5
13251519.3
13259519.9
13283521.8
13287522.1
13319524.5
17120.1
171711.0
1723181.4
19550.3
191911.0
23520.2
23720.3
232311.0
2341261.8
25510.2
25780.3
251740.7
252511.0
259533.8
2511534.6
259351737.4
2517351769.4
29110.0
291190.4
292911.0
29381141131.4
29705541243.3
3113120.4
313111.0
35710.2
351340.4
351740.5
352520.7
353511.0
3513333.8
3516134.6
3513091737.4
3524291769.4
371930.5
372330.6
372930.8
373711.0
41180.0
414111.0
43130.0
434311.0
47570.1
4725160.5
474711.0
476541.4
477341.6
478541.8
478941.9
4710142.1
4925220.5
493520.7
494911.0
535311.0
53103171.9
55140.0
55520.1
55720.1
551110.2
5541160.7
555511.0
556581.2
5520933.8
5525334.6
5520571737.4
5538171769.4
591110.0
595911.0
5913362.3
5914962.5
5918163.1
5918563.1
5921763.7
5922163.7
61110.0
616111.0
61235413.9
65510.1
651310.2
6519120.3
656511.0
6524733.8
6529934.6
656551510.1
651055516.2
651135517.5
651255519.3
651295519.9
651415521.8
651435522.1
651595524.5
6524311737.4
6545111769.4
6717160.3
676711.0
712950.4
713150.4
717111.0
7125851736.4
7128091739.6
7139851756.1
7141211758.0
7144091762.1
73540.1
7319320.3
734780.6
737311.0
771160.0
77720.1
775520.7
777711.0
779181.2
791200.0
797200.1
797911.0
79233142.9
79265143.4
8365140.8
838311.0
8310971.3
8315771.9
85520.1
857560.1
851710.2
858511.0
85115181.4
8532333.8
8539134.6
8531791737.4
8558991769.4
891780.2
898911.0
911240.0
91710.1
912560.3
915960.6
916520.7
919111.0
919171510.1
911477516.2
911589517.5
911757519.3
911813519.9
911981521.8
912009522.1
912233524.5
951360.0
951750.2
951910.2
952350.2
952550.3
959511.0
9536133.8
9543734.6
9535531737.4
9565931769.4
97160.0
971330.1
979711.0
101760.1
1011160.1
1011930.2
1012330.2
1012930.3
1013130.3
1013730.4
10110111.0
103580.0
10323220.2
10310311.0
KSeedSizeSeed/K
1071530.0
10710711.0
10919300.2
10910911.0
1131120.0
11311311.0
11513130.1
11517180.1
1152310.2
1152520.2
1153520.3
11511511.0
115205261.8
11543733.8
11552934.6
11543011737.4
11579811769.4
1191420.0
119520.0
119720.1
1191120.1
11923180.2
1198520.7
11911911.0
119125141.1
119161181.4
1215180.0
1211120.1
12119460.2
12112111.0
12114381.2
125110.0
1252510.2
1253580.3
12547190.4
1258540.7
12512511.0
125143191.1
12547533.8
12557534.6
125899747.2
12546751737.4
12586751769.4
1271180.0
12741180.3
12712711.0
13113130.1
13117130.1
13123260.2
13113111.0
13311180.1
1333550.3
13359240.4
1339520.7
13313311.0
137140.0
13741400.3
13713711.0
137503183.7
137743185.4
137967187.1
13911740.1
13913911.0
1437800.0
1431110.1
1431320.1
1431770.1
1432970.2
14314311.0
14316981.2
14314411510.1
1432321516.2
1432497517.5
1432761519.3
1432849519.9
1433113521.8
1433157522.1
1433509524.5
145120.0
145510.0
14523100.2
1452910.2
14547180.3
1455590.4
14514511.0
14555133.8
14566734.6
14554231737.4
145100631769.4
1451905541131.4
1453527541243.3
14919330.1
14914911.0
149667264.5
15141400.3
151113100.7
15115111.0
1551180.0
1553110.2
15565120.4
15515511.0
15558933.8
15571334.6
15557971737.4
155107571769.4
15713120.1
15715711.0
1611160.1
1611960.1
16125220.2
1613520.2
1614920.3
16179160.5
16111520.7
16116111.0
161287261.8
16311490.1
16317360.1
16316311.0
16713200.1
1679590.6
16712790.8
16716711.0
1691190.1
1691310.1
16917230.1
16916911.0
16917031510.1
1692743516.2
1692951517.5
1693263519.3
1693367519.9
1693679521.8
1693731522.1
1694147524.5
1737140.0
1733770.2
17317311.0
17529120.2
1753510.2
1754980.3
1755380.3
1756540.4
1757340.4
1758540.5
1758940.5
17510140.6
17510340.6
17511940.7
17512520.7
17514340.8
17517511.0
17566533.8
17580534.6
17565451737.4
175121451769.4
1795620.0
17917911.0
18111400.1
18123100.1
18155100.3
18118111.0
1851360.0
1853710.2
1859530.5
18511530.6
18514530.8
18518511.0
18570333.8
18585134.6
18569191737.4
185128391769.4
1875120.0
1871120.1
1871720.1
18743120.2
18718711.0
18722181.2
187253181.4
19147240.2
19119111.0
191961255.0
19124052512.6
1931290.0
193550.0
19331120.2
19391500.5
19319311.0
1975690.0
19719711.0
1991360.1
1994760.2
19919911.0

Largest Seed

Searching k values up to 199 found a 41 member loop in 3x+145. This loop had the largest seed of those discovered in this search.

35,275  52,985  39,775  59,735  89,675  134,585  100,975
151,535  227,375  341,135  511,775  767,735  1,151,675  1,727,585
1,295,725  485,915  728,945  546,745  410,095  615,215  922,895
1,384,415  2,076,695  3,115,115  4,672,745  3,504,595  5,256,965  492,845
184,835  277,325  104,015  156,095  234,215  351,395  527,165
197,705  148,315  222,545  166,945  125,245  46,985  35,275

145 and 35275 have a common factor (5), so there is also a 41 member loop in 3x+29 (145/5) which has the seed 'S' based on 7055 (35275/5). These are essentially the same loop and have the same S/K value, see the next section.

Record Holders

In an earlier section we observed the value of s/k for systems up to 3x+199. We also noticed that the largest loop seed found was in the 3x+145 system. If we look at the more interesting s/k value (it gives the same value for the essentially identical loops found in 3x+29 and 3x+145) we find the largest value so far is in 3x+29.

SystemSeedSizeSeed/K
3x+1111.0
3x+53471769.4
3x+29705541243.3

This particular record isn't an easy one to calculate. To produce the above values I had to run 100 million calculations for 67 different systems. Even then I don't know if I've missed any loops, there may be a ridiculously large loop buried somewhere in 3x+1, for instance. It does look though that any loops there are are either close to 'k' or very large. I've not found any intermediate sized loops.

One way of scanning large numbers of systems would be to just choose a few numbers and see where they end up. '1' might be a good starting point and we can scan a large number of systems quite quickly. The corresponding results are.

SystemSeedSizeSeed/K
3x+1111.0
3x+2341261.8
3x+53103171.9
3x+191961255.0

This hasn't worked very well, possibly because a seed of 1 tends to find the smaller loops. How about trying a big number, say 1000001? This finds the 131 loop in 3x+13 giving a slightly higher record of 10.1, but still nowhere near the real record of 243.3.

Convergence Distribution

There don't seem to be any shortcuts for finding the loop record holders. Maybe this is because the majority of seeds converge on one of the smaller loops. Lest test this hypothesis in the 3x+29 system which has 5 loops including the record holder based on 7055.

First let's try all the seeds less than 7055. The results are shown in the table on the left. The table on the right shows the same test using the first 10 million odd numbers. Only about 0.28% converge on 7055.

Loop Frequency
1 291
11 3101
29 122
3811 10
7055 3
Loop Frequency
1 794171
11 8793106
29 344828
3811 39628
7055 28267

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(c) John Whitehouse 2011 - 2017