The Multiplicative Persistence Conjecture: Resolving the 2-Adic Obstruction for Nonzero Even Targets
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,…,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≥3 and to every prime p| 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.
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BibTeX
@misc{fonga2026multiplicative,
title = {{The Multiplicative Persistence Conjecture: Resolving the \(2\)-Adic Obstruction for Nonzero Even Targets}},
author = {Patrick Nyadjo Fonga},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27802},
eprint = {2608.27802},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27802},
note = {CC0},
}
RIS
TY - UNPB
TI - The Multiplicative Persistence Conjecture: Resolving the 2-Adic Obstruction for Nonzero Even Targets
AU - Patrick Nyadjo Fonga
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.27802
UR - https://arxiv.org/abs/2608.27802
AB - 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,…,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≥3 and to every prime p| 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.
ER -