The Multiplicative Persistence Conjecture: Resolving the $2$-Adic Obstruction for Nonzero Even Targets

Source abstract

The multiplicative persistence problem studies the process of repeatedly replacing a positive integer by the product of its digits until a single digit, called the terminal digit, is reached. The classical conjecture asserts that no decimal integer requires more than $11$ iterations. Brier, Clavier, Gutsche, and Naccache proved the conjecture for all odd terminal digits. In their approach to nonzero even terminal digits, they were led to infinite families of decimal integers in which the numbers of digits $2,\ldots,9$ are fixed, while arbitrarily many digits $1$ may be inserted. They conjectured that, despite the infinitude of such a family, the exponent of $2$ dividing its elements is uniformly bounded. We prove this conjecture and obtain an explicit bound depending only on the prescribed digit multiplicities. More generally, our argument applies in every base $b\geq3$ and to every prime $p\mid b$. In base $10$, combining our bound with the method of Brier, Clavier, Gutsche, and Naccache yields a finite algorithm for a further analysis of each nonzero even terminal digit.