On the Collatz conjecture (Q2868719)

Starting out from an odd positive integer \(n\) define the Collatz sequence \((a_{0}, a_{1}, \dots{})\) with the following recursion: \(a_{0} = n\), \(a_{k+1} = F(a_{k})\), where \(k \geq 0\) and \(F(m)\) is the speeded-up Collatz function. Let \(r_{k}\) be the maximal exponent such that \(2^{r_{k}}\) divides \(3a_{k}+ 1\). The author conjectures that for every positive odd integer \(n\) there exists \(k \geq 0\) integer such that \(\lceil k\log_{2}3 \rceil \leq \sum_{j = 0}^{k-1}r_{j}\). For \(k \geq 1\) let \(R_{0}, \dots{}, R_{k-1}\) be positive integers and let \(A_{k} = \sum_{j = 0}^{k-1}R_{j}\). The author also conjectures that if \(A_{l} < \lceil l\log_{2}3 \rceil\) for every \(1 \leq l \leq k - 1\) and \(A_{k} = \lceil k\log_{2}3 \rceil\) then the minimal solution \((x_{0}, y_{0})\) of the equation \(2^{A_{k}}y - 3^{k}x = \sum_{i=1}^{k}3^{k-i}2^{A_{i-1}}\) in positive integers satisfies \(y_{0} < x_{0}\) or \(x_{0} = 1\). He proves that the above conjectures imply the Collatz conjecture. The proof uses elementary observations. The paper contains numerical results which confirm the second conjecture.