Under the unproven Weak Residue Conjecture, the paper derives six logarithm formulas linking total, odd, and even step counts of a Collatz trajectory, one of which is equivalent to the conjecture itself.
The relationship between stopping time and number of odd terms in Collatz sequences
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abstract
The Collatz sequence for a given natural number $N$ is generated by repeatedly applying the map $N$ $\rightarrow$ $3N+1$ if $N$ is odd and $N$ $\rightarrow$ $N/2$ if $N$ is even. One elusive open problem in Mathematics is whether all such sequences end in 1 (Collatz conjecture), the alternative being the possibility of cycles or of unbounded sequences. In this paper, we present a formula relating the stopping time and the number of odd terms in a Collatz sequence, obtained numerically and tested for all numbers up to $10^7$ and for random numbers up to $2^{128.000}$. This result is presented as a conjecture, and with the hope that it could be useful for constructing a proof of the Collatz conjecture.
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Iteration Steps of 3x+1 Problem
Under the unproven Weak Residue Conjecture, the paper derives six logarithm formulas linking total, odd, and even step counts of a Collatz trajectory, one of which is equivalent to the conjecture itself.