The Multiplicative Persistence Conjecture Is True for Odd Targets

TL;DR

This paper proves the multiplicative persistence conjecture for all odd target values, showing that the number of steps to reach the digital root is bounded for these cases, advancing understanding of this longstanding number theory problem.

Contribution

It establishes the truth of the multiplicative persistence conjecture specifically for odd target values, providing bounds on the steps needed to reach the digital root.

Findings

Proves the conjecture for odd targets: 1, 3, 7, 9 with at most 1 step.

Shows for target 5, the persistence is at most 5 steps.

Discusses challenges in extending results to even 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.