Explorations of the Collatz Conjecture - Moravian College

3.1 Applying Garner’s Stems
In this section we show that if an integer m has a parity vector beginning with s i ,
then the integer m + 1 must have a parity vector beginning with s ′ i
. The following
lemma and theorem combine to give this result.
Lemma 3. If P i+1 (m) = 〈0 11...1〉, then P i+1 (m + 1) = 〈1 11...1〉 and T i+1 (m + 1) =
3T i+1 (m) + 2.
}{{}
i 1’s
}{{}
i 1’s
Proof. We proceed by induction. Base case: i = 0
Assume P 1 (m) = 〈0〉. Then we know that m ≡ 0 mod 2. This means that m + 1 ≡
1 mod 2. So, P 1 (m + 1) = 〈1〉. Therefore the statement is true for i = 0.
3.1,
Assume the statement is true for i = n − 1.
Next, suppose i = n. Then, we are assuming T n (m) ≡ 1 mod 2. From Equation
T n (m) = 3n−1
2 (m) + 1 ∑n−2
( ) n−i 1 n 2 + 3 n−i−1
2
i=1
= 1 ( ) n 3
m + 1 3 2 2 + 1 ∑n−2
( ) n−i 3
.
3 2
Since we are assuming the statement is true for i = n−1, P n (m+1) = 〈1 11...1〉.
i=1
}{{}
n−1 1’s
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