Theorem C asserts that the set of integers with a given coefficient
stopping time k is a set of arithmetic progressions
, which has the immediate consequence that has the
which is given by
Furthermore Theorem C asserts that the set
differs from by a finite set, so that also has an asymptotic density
which is the same as that of .
Consequently, Theorem C implies the first part of Theorem A, that the set
of all integers with stopping time at most k have an asymptotic density
Now the formula (2.16) can be used to prove the second part of Theorem A,
and in fact to prove the stronger result that approaches 1 at an exponential rate as .