A terminating, non-cyclic* path towards the Collatz conjecture

Jon Seymour <[email protected]>

14 October 2023 (updated: 22 October 2023)

Abstract

"Mathematics may not be ready for such problems" - Paul Erdős on the Collatz conjecture.

It has long been known that the following identity applies to all paths of the Collatz sequence.

$2^{e}x_{n} - 3^{o}x_{0} = k$

where $x_{0}$ is the initial term and $x_{n}$ is the final term and $e$, $o$ and $k$ are path dependent parameters.

This paper derives a formula for k derived from 3 parameter sequences $m_{n}, o_{n}, e_{n}$.

$m_n = x_n\pmod{2}$

$o_{n} = \sum_{k=0}^{k=n}{m_k}$

$e_{n} = n+1 - o_{n}$

$k_{n} = 2^{e_{n-1}}x_{n}-3^{o_{n-1}}x_0=\sum_{i=0}^{i=n-1}{{2^{e_{i}}3^{o_{n-1}-o_i}m_{i}}}$

The resultant identity is not novel, for example see [1], but perhaps the technique by which it was derived may be interesting to some.

We also derive a recurrence relation that expresses each $k_{n}$ in terms of $k_{n-1}$

$k_n = 3^{m_{n-1}}k_{n-1}+2^{e_{n-1}}m_{n-1}$

and show that $k_{0} = 0$

[1] First odd term of the sequence lower odd number $n$ related to the $3\cdot n+1$ on Maths Overflow Net

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