3x+1 Strength Records

The Strength S of a number N is defined as S(N) = 5.O - 3.E, where O and E represent the number of Odd (3x+1) and Even (x/2) steps respectively.

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 < 2100. Even the smallest of these, the candidate for record number six, is so large we may never be able to confirm it unless a totally new approach can be found.

(Potential) Strength records currently known < 2100
# Strength Delay Number
11? 68 3820 35439,711767,561429,622628,637695
10? 65 3661 2421,645885,513162,728286,680347
9? 57 3429 108,123012,240928,276929,135463
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

Back to the general 3x+1 page.