The Multiplicative Persistence Conjecture Is True for Odd Targets

Abstract

In 1973, Neil Sloane published a very short paper introducing an intriguing problem: Pick a decimal integer $n$ and multiply all its digits by each other. Repeat the process until a single digit $\Delta(n)$ is obtained. $\Delta(n)$ is called the \textsl{multiplicative digital root of $n$} or \textsl{the target of $n$}. The number of steps $\Xi(n)$ needed to reach $\Delta(n)$, called the multiplicative persistence of $n$ or \textsl{the height of $n$} is conjectured to always be at most $11$. Like many other very simple to state number-theoretic conjectures, the multiplicative persistence mystery resisted numerous explanation attempts. This paper proves that the conjecture holds for all odd target values: Namely that if $\Delta(n)\in\{1,3,7,9\}$, then $\Xi(n) \leq 1$ and that if $\Delta(n)=5$, then $\Xi(n) \leq 5$. Naturally, we overview the difficulties currently preventing us from extending the approach to (nonzero) even targets.