MrHellmet
- 2
- 0
- TL;DR Summary
- I am empirically studying the transitions between successive odd numbers in the Collatz sequence, focusing on the change in the 2-adic valuation (Δv) between steps. I've observed that integers with a constant Δv form arithmetic progressions.
Hello everyone,I've been empirically analyzing the transitions between successive odd numbers in the Collatz sequence (m_{new} = (3m_{old}+1)/2^N).I've observed that odd integers that produce a specific "valuation jump", \Delta v = v_2(3m_{new}+1) - v_2(3m_{old}+1), seem to fall into arithmetic progressions. I've been studying the 'seed' of these progressions, defined as the smallest positive odd integer m satisfying the conditions for a given family.For the family of integers with an initial valuation v_2(3m+1) = 2, the seeds for positive valuation jumps (\Delta v = k \geq 1) appear to follow this recursive formula:
a(k) = a(k-1) + C_{pos}(2,k) \cdot 2^{k+3}
Here, a(k) is the seed for a jump of \Delta v = k, and the starting seed is a(1)=17. The coefficient C_{pos}(2,k) is periodic depending on
k \pmod 6:C_{pos}(2,k) = \begin{cases} 3 & \text{if } k \equiv 3 \pmod 6 \\ -1 & \text{if } k \equiv 0 \pmod 6 \\ 1 & \text{otherwise} \end{cases}
For example, this correctly predicts the seed for \Delta v = 2:a(2) = a(1) + C_{pos}(2,2) \cdot 2^{2+3} = 17 + (1) \cdot 32 = 49.My questions are:1. Is this specific recursive structure, or the periodic nature of its coefficient, a known result in the literature on the 3n+1 problem?2. If it is not a known result, does anyone see a potential path or a related mathematical structure that could help in proving it?I have verified this and similar patterns for other families extensively, but I haven't been able to find a formal proof. Any references or insights would be greatly appreciated.
a(k) = a(k-1) + C_{pos}(2,k) \cdot 2^{k+3}
Here, a(k) is the seed for a jump of \Delta v = k, and the starting seed is a(1)=17. The coefficient C_{pos}(2,k) is periodic depending on
k \pmod 6:C_{pos}(2,k) = \begin{cases} 3 & \text{if } k \equiv 3 \pmod 6 \\ -1 & \text{if } k \equiv 0 \pmod 6 \\ 1 & \text{otherwise} \end{cases}
For example, this correctly predicts the seed for \Delta v = 2:a(2) = a(1) + C_{pos}(2,2) \cdot 2^{2+3} = 17 + (1) \cdot 32 = 49.My questions are:1. Is this specific recursive structure, or the periodic nature of its coefficient, a known result in the literature on the 3n+1 problem?2. If it is not a known result, does anyone see a potential path or a related mathematical structure that could help in proving it?I have verified this and similar patterns for other families extensively, but I haven't been able to find a formal proof. Any references or insights would be greatly appreciated.
Last edited: