Explorations of the Collatz Conjecture - Moravian College

We can use Lemma 7, Equation 3.1, Theorem 6 and our definitions of blocks
and strings to come up with a new formula for checking to see if a set of parity
sequences is in fact a block or string prefix.
Theorem 7. If v and v ′ are parity sequences with n ones and k entries, they are
block or string prefixes if and only if
n∑
2 k − 3 n 3 n−t
= δ t
2 k−a+1 (2r − 1).
t=1
Proof. First, assume v = 〈v 1 , v 2 , ..., v k−1 , v k 〉 and v ′ = 〈v ′ 1 , v′ 2 , ..., v′ k−1 , v′ k
〉 are block
prefixes. Then, by definition, we know that T v (x) + 1 = T v ′(x + 1). From Equation
3.1 and Theorem 6 we know that when v ′ is applied to x + 1 the result is
∑
T v ′(x + 1) = 3n
k−1
2 (x + 1) + k
i=0
3 (v′ k +v′ k−1 +...+v′ i+2 )
v ′
2 k−i i+1 + = T 3n
v ′(x) +
2 . k
Therefore, the following should be true: T v (x) + 1 = T v ′(x) + 3n
2 k . Using the substitution
of T v ′(x) = T v (x) + ∆ from Lemma 7,
T v (x) + 1 = T v (x) + ∆ + 3n
2 k
∆ = 1 − 3n
2 k
2 k ∆ = 2 k − 3 n .
From Lemma 7, 2 k ∆ = ∑ n 3
t=1 δ n−t
t (2 r − 1). Therefore,
2 k−a+1 n∑
2 k − 3 n 3 n−t
= δ t
2 k−a+1 (2r − 1).
t=1
A similar proof can be written for the case where v and v ′ are string prefixes.
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