Explorations of the Collatz Conjecture - Moravian College

Chapter 5
Long Runs of Consecutive Integers
Recall from Chapter 1 that we define a run to be a set of consecutive integers that
all have the same total stopping time. Garner [4] pointed out that runs are simply
many sets of overlapping pairs of consecutive integers of the same total stopping
time. This suggests a way to build long runs. First, start with a run, then try to
make the last integer of the run the first part of a pair. If this can be done, then we
have found a new run with at least one more consecutive integer than our starting
run. Using this idea, we feel that given any starting parity sequence of length k,
we can construct an arbitrarily long run where that parity sequence is the first k
terms of the parity vector of the first integer in the run. The following conjecture
states this idea more formally.
Conjecture 3. Consider any parity vector of length k, which corresponds to the
set of integers of the form 2 k m + b k , where b k is found using Terras’[6] algorithm,
m is some positive integer, and σ ∞ (2 k m+b k ) = σ ∞ (2 k m+b k +1) = ... = σ ∞ (2 k m+
b k +(g−1)) (a run of g consecutive integers of the same total stopping time.) There
exists some positive integer m ′ such that σ ∞ (2 k m ′ + b k ) = σ ∞ (2 k m ′ + b k + g), a run
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