  The 3N+1-problem, also known as The Collatz Conjecture, concerns the
behaviour of natural numbers N when the following rule is used repeatedly: If N
is odd then multiply N by three and add one (a type Odd operation/ iteration),
if N is even divide N by two (a type Even operation/ iteration).
  The Collatz Conjecture states that for any N as start-value the
iteration-series must eventually reach the known Loop -4-2-1-4-... In the
present work is shown that all iteration-series reach a value lower than the
start-value, N > 1, at some point in the iteration-series, which is equivalent
to showing that all iteration-series also reaches the known Loop. This is done
by analysing the behaviour of subsets of the natural numbers. The subsets are
called classes: (AX-B), B < A, where the independent variable X is a positive
integer, the modulo A is a positive integer and the constant B is a
non-negative integer. It is shown that there exist a countable infinity of
"reducing" classes e.g. all elements in the class (2X) = {2,4,6,8,10,...} are
reduced in one type Even operation and all elements in the class (4X-3) =
{1,5,9,13,17,...} are reduced in one type Odd operation followed by two type
Even operations.
  A formula is deduced that for any number of type Odd operations, #O = s,
delivers the exact number of type Even operations, #E = r, required to obtain a
"reducing combination" of operations: r = ceiling(sZ), s > 0, Z = log3/log2 =
1,5849625007...
  Another formula is deduced that for any "reducing combination"
(s,ceiling(sZ)), s > 1, delivers the exact number of "reducing" classes having
modulo A = 2^ceiling(sZ). By the help of a graph in form of an infinite
complete binary tree it is then shown that all natural numbers N must belong to
exactly one "reducing" class, thus proving that The Collatz Conjecture is true.
