# inequality – Collatz Conjecture: Are the following valid properties of a cycle?

Let:

• $$nu_2(x)$$ be the 2-adic valuation of $$x$$
• $$x_1, x_2, dots, x_k$$ be distinct odd integers that make up a cycle of length $$k$$ with:
• $$x_{i+1} = dfrac{3x_i+1}{2^{nu_2(3x_i+1)}}$$
• $$x_{i+k} = x_i$$
• $$p_1, p_2, dots, p_k$$ be positive integers associated with the above cycle in the following way:
• Each $$p_i = sumlimits_{t=1}^i nu_2(3x_t+1)$$
• $$p_k > p_{k-1} > dots > p_2 > p_1 > 0$$
• $$2^{p_k} > 3^k$$
• $$2^{p_k}x_{k+1} – 3^{k}x_1 = 3^{k-1} + sumlimits_{m=1}^{k-1}3^{k-1-m}2^{p_m}$$

Note: Details for this last equation can be found here.

• $$s_1, s_2, dots, s_{k+1}$$ be positive integers that are not a cycle but are also characterized by $$p_1, p_2, dots, p_k$$ so that:

• $$s_1 < x_1$$
• $$s_{i+1} = dfrac{3s_i+1}{2^{nu_2(3s_i+1)}}$$
• $$s_{k+1} ne s_1$$
• $$2^{p_k}s_{k+1} – 3^{k}s_1 = 3^{k-1} + sumlimits_{m=1}^{k-1}3^{k-1-m}2^{p_m}$$
• $$t_1, t_2, dots, t_{k+1}$$ be positive integers that are not a cycle but are also characterized by $$p_1, p_2, dots, p_k$$ so that:

• $$t_1 > x_1$$
• $$t_{i+1} = dfrac{3t_i+1}{2^{nu_2(3t_i+1)}}$$
• $$t_{k+1} ne t_1$$
• $$2^{p_k}t_{k+1} – 3^{k}t_1 = 2^{p_k}s_{k+1} – 3^{k}s_1 = 2^{p_k}x_{k+1} – 3^{k}x_1$$

Question:

In the case of the trivial cycles, $$s_1, dots, s_{k+1}$$ may not exist. Since there are an infinite number of non-cycles for any combination of $$p_1, dots, p_k$$, there are always an infinite number of instances of $$t_1, dots, t_{k+1}$$

It seems to me that if a cycle exists, then for each $$s_1, dots, s_{k+1}$$ that exists:
$$s_{k+1} > s_1$$

For each $$t_1, dots, t_{k+1}$$ that exists
$$t_1 > t_{k+1}$$

Am I right? Did I make a mistake? Is any point in the question unclear?

Here’s the argument:

(1) $$2^{p_k}(x_{k+1} – s_{k+1}) = 3^k(x_1 – s_1)$$

(2) Since $$2^{p_k} > 3_{k}$$, it follows that:

$$x_{k+1} – s_{k+1} < x_1 – s_1$$

$$s_1 – s_{k+1} < x_1 – x_{k+1} = 0$$

(3) $$2^{p_k}(t_{k+1} – x_{k+1}) = 3^k(t_1 – x_1)$$

(4) So:

$$t_{k+1} – x_{k+1} < t_1 – x_1$$

$$t_{k+1} – t_1 < x_{k+1} – x_1 = 0$$