A Fractional $3n+1$ Conjecture

Abstract

In this paper we introduce and discuss the sequence of \emph{real numbers} defined as $u_0 \in \mathbb R$ and $u_{n+1} = \Delta(u_n)$ where \begin{equation*} \Delta(x) = \begin{cases} \frac{x}{2} &\text{if } \operatorname{frac}(x)<\frac{1}{2} \\[4px] \frac{3x+1}{2} & \text{if } \operatorname{frac}(x)\geq\frac{1}{2} \end{cases} \end{equation*} This sequence is reminiscent of the famous Collatz sequence, and seems to exhibit an interesting behaviour. Indeed, we conjecture that iterating $\Delta$ will eventually either converge to zero, or loop over sequences of real numbers with integer parts $1,2,4,7,11,18,9,4,7,3,5,9,4,7,11,18,9,4,7,3,6,3,1,2,4,7,3,6,3$. We prove this conjecture for $u_0 \in [0, 100]$. Extending the proof to larger fixed values seems to be a matter of computing power. The authors pledge to offer a reward to the first person who proves or refutes the conjecture completely -- with a proof published in a serious refereed mathematical conference or journal.