2021/10/08 by Eric Brier, Christophe Clavier, Brier, Eric +5
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.2110.04263
arxiv created 2021/10/08 · arxiv updated 2021/10/11
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 Δ(n) is obtained. Δ(n) is called the multiplicative digital root of n or the target of n. The number of steps Ξ(n) needed to reach Δ(n), called the multiplicative persistence of n or 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 Δ(n)∈\1,3,7,9\, then Ξ(n) ≤ 1 and that if Δ(n)=5, then Ξ(n) ≤ 5. Naturally, we overview the difficulties currently preventing us from extending the approach to (nonzero) even targets.