3x+1 Strength and Strength Records
Note that the Strength can be defined equally well when the algorithm is defined using (3x+1)/2 rather than 3x+1. In that case again let O' and E' be the number of Odd and Even iterations, and we have S(N) = 2.O' - 3.E'. This makes the Strength independent of the choice of algorithm.
It should be obvious that any Strength record is necessarily a Class Record as well. But the Strength turns out to be fairly constant over large ranges of Class Records. So much so that only a few Strength records are known. Current data suggest they are roughly logarithmically distributed and occur only once in every ten powers of two or thereabout.
Depending on the exact definition of the 3x+1 function the first Strength record is either 1 or 2. The number N=1 has a Delay of zero, and therefore also Strength zero. If one does not want to see 1 as the first record the first record is 2, with a S(2) = -3. Since either number is rather trivial the first Strength record in the table below has been labeled as #0. It is rather surprising to see that neither value is surpassed by any N < 60,000,000.
As the table shows the first non-trivial Strength record has 8 digits already, thus
emphasizing their rarity. For many years only four Strength records were known
with certainty. In 2011 the results of the distributed search
eventually confirmed Strength record number five.
The other numbers depicted are current best known candidates for the next records.
At this moment these are the only candidates known for N < 2^{100}.
The smallest of these, the candidate for record number six, may just be reachable with
the current distributed project, but the next candidates are so large we may never be
able to confirm them unless a totally new approach can be found.
(Potential) Strength records currently known < 2^{100} | |||
# | Strength | Delay | Number |
13? | 70 | 3918 | 181757,895437,676542,787284,912127 |
12? | 69 | 3825 | 31549,861135,742690,018597,749695 |
11? | 65 | 3661 | 2421,645885,513162,728286,680347 |
10? | 60 | 3316 | 4,783763,851719,920612,626591 |
9? | 58 | 3154 | 239958,815015,359848,833791 |
8? | 56 | 2968 | 7219,136416,377236,271195 |
7? | 55 | 2955 | 6852,539645,233079,741799 |
6? | 46 | 2710 | 268,360655,214719,480367 |
5 | 38 | 2254 | 104899,295810,901231 |
4 | 28 | 1820 | 100,759293,214567 |
3 | 17 | 1549 | 3,743559,068799 |
2 | 10 | 1210 | 13371,194527 |
1 | 9 | 949 | 63,728127 |
0 | 0 | 0 | 1 |
Therefore numbers with Strength -5; -6; -7 and so on are simply 3; 4; 5; 6; 8; 10; 12; 16; 20; 24; 32; 40 and so on.
For S ≥ -4 however the situation is far less obvious. For instance, the smallest numbers with Strengths 0; -1; -2; -3 and -4 are 1; 209,303017; 235,465895; 2; and 179,008815 .
There may exist numbers with very high Strengths though, since the list of Class Records shows slowly increasing Strength values. Indeed, it appears that for every Strength S there is a limit beyond which all Class Records have higher Strengths. The evidence is rather thin, but the table below depicts the highest Class Records with Strength S.
Highest Known Class Records with particular Strength (>= -8) | |||
Strength | Delay | Number | |
-8 | 920 | 72945,377791 | |
-7 | 1237 | 1,024337,865708 | |
-6 | 1274 | 1,726869,407806 | |
-5 | 1375 | 11,787870,157403 | |
-4 | 1484 | 95,427878,556519 | |
-3 | 1473 | 60,387954,399047 | |
-2 | 1486 | 63,618585,704351 | |
-1 | 1523 | 112,265328,502742 | |
0 | 1528 | 100,197749,947111 | |
1 | 1525 | 74,843552,335161 | |
2 | 1698 | 2421,269892,238985 | |
3 | 1719 | 2933,495759,934703 | |
4 | 1724 | 2687,265553,316295 | |
5 | 1721 | 2034,106312,229086 | |
6 | 1742 | 2510,719179,521742 | |
7 | 1739 | 1883,039384,641307 | |
8 | 1776 | 3275,533932,696303 | |
9 | 1925 | 60614,485209,289967 | |
10 | 1882 | 20089,765945,518247 | |
11 | 1951 | 69520,368888,759751 | |
12 | 2076 | 766624,412109,824743 | |
13 | 2113 | 1,345161,942161,756487 | |
14 | 2142 | 2,029435,401161,373768 | |
15 | 2131 | 1,284252,089797,431844 | |
16 | 2192 | 3,749740,882243,642695 | |
17 | 2165 | 1,694294,236575,537470 | |
18 | 2234 | 5,780540,886094,271145 | |
19 | 2319? | 28,424489,287394,211294 |
Thus we may venture
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