The graph on the right depicts ²log(Mx(P_i )) against ²log(P_i ) for all known Path Records. The tendency to coincide with the white reference line which has a slope of exactly 2.0 is striking.

The table below contains the first 87 Path Records as currently found or confirmed by the author. These results exactly match those found by Tomás Oliveira e Silva who earlier determined all Path Records up to 100.2⁵⁰ and in 2008 extended his search to 5.2⁶⁰. During this last search he found Path Record 88.

In 2017 the yoyo@home project searched the interval up to 87.2⁶⁰. They were able to confirm the higher records (from #76 onwards) and also found four new ones.

As of 2019 David Bařina is expanding the convergence range. So far the search has also produced four more Path Records.

In the table the first column depicts the record number. N is the Record, Mx(N) is the maximum value reached. X₂(N) is equal to the Expansion, Mx(N) / N². The five known Expansion records have been indicated with a different color.

The next two columns represent the number of bits needed to store N and Mx(N) respectively. The number of bits needed to store any number x is obviously equal to [ ²log(x) ] + 1. The last column gives the author who first found or published the record. For the lowest records this is obviously a trivial affair, therefore this column is left blank for all numbers of 32 bits or less.

From the values in the sixth column it is simple to determine the number of bits one needs when calculating 3x+1 paths up to a certain number. Note that one less bit can be used by calculating a multiplication step and the division by 2 immediately following it by using x + [x/2] + 1 rather than simply 3x + 1 followed by a division by 2.

Even without that last refinement it is interesting to see that the complete paths of all numbers up to 8 bits can be calculated in 16 bits, and likewise all numbers up to 16 bits take only 32 bits and so on for every multiple of 8 bits. Although we encounter several cases where X₂(N) is larger than 1 it so happens that none of these cases occur "just below" a power of 256 (2⁸). The table therefore establishes the practical fact that for all numbers in the interval researched so far the path of every number taking k bytes (assuming a byte consists of 8 bits) can be completely determined using a storage of just 2k bytes for intermediate results. Or, stated more accurately:

Observation :

For all positive numbers N  <   2⁷⁰ : [²⁵⁶log( Mx(N) ) + 1]   ≤  2 . [²⁵⁶log(N) + 1].