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Paper 2

A discussion of the implications of the findings in Paper 1
1 If, instead of using 3x+1, one uses 3x-1, one soon finds integers which do not converge to 1.

In fact there are three known graphs for 3x-1:

  1. a graph that converges to 1
  2. a graph loops on the thread 5>>7>>5>>7>> . . . .
  3. a graph loops on the thread 17>>25>>37>>55>>41>>61>>91>>17>> . . . .

If a similar algorithm for even integers is used as before, then, for 3x-1, 7==2, 55 == 14 and 91==23==6

The least members in the three trees are respectively 1, 2 and 6; they could hardly be smaller.
2 The ratios and equations for nchains for 3x-1 parallel those for 3x+1, i.e. the ratio of 13/16 appears to apply equally well to 3x-1 as it does to 3x+1.

Why should 3x-1 loop on integers other than 1 while 3x+1 only loops on 1?
3 Replacing 3x+1 with 3x-1 is exactly equivalent to using negative integers as start values and using 3x+1.

When using 3x-1 and positive start values, all equation signs are reversed, e.g. the 4p+1 rule becomes the 4p-1 rule, the 2chain link cycle becomes an alternation of 8p-3 and 8p-4, etc.
Examples

using 3x+1 and starting with -1 -1 >> -2 >> -1
using 3x-1 and starting with +1 1 >> 2 >> 1
using 3x+1 and starting with -5 -5 >> -7 >> -5
using 3x-1 and starting with +1 5 >> 7 >> 5
using 3x+1 and starting with -17 -17 >> -25 >> -37 >> -55 >> -41 >> -61 >> -91 >> -17 >> . . . .
using 3x-1 and starting with +17 17 >> 25 >> 37 >> 55 >> 41 >> 61 >> 91 >> 17 >> . . . .
4 Investigating the graphs that exist if other equations are used, e.g. 5x+1, 7x+1, etc., has shown that in every case where two or more graphs exist, there are comparatively small integers as least members in such graphs, which are very soon found.
5 It seems that there ought to be a way of using the following facts to prove that, if more than one graph exists, then the least value in each graph must be comparatively small and of similar magnitude, i.e. the graphs exist in parallel right from the start, rather a new graph suddenly appearing, consisting of extremely large values.

  1. that a graph has an infinite number of levels
  2. that when constructing the graphe which converges to 1, the integers available to construct the nchain at Ln and the infinity of nchains at higher levels than Ln, decreases by a factor of 13/16 for every increment of n
  3. that no single integer can fail Collatz as every integer must be part of an infinitely large graph of integers
  4. that, for n = 100, a comparatively small value, the integers available to construct the 100chain at L100 are less than 1 in a billion
  5. that the search for a failing value has reached astronomical values without success
  6. every 3x+1 graph, if more than one exists, must contain nchains constructed in exactly the same way as those in the graphe converging to 1.
  7. a graph which does not loop is not feasible; in such a graph every level would consist of nchains for every value of n from 1 to infinity.
6 Although (13/16)n is infinitely small as n tends to infinity, an infinitely small part of infinity is still an infinitely large number. The extensive use of the tables in the Appendices to consistently produce the required chains in Paper 3, is a confirmation of their accuracy.

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