An elementary note on a digit–product condition forcing repunits
Year: 2025
Type: article
Abstract: We study a very simple condition on the decimal digits of a positive integer. Let PD(n) be the product of the decimal digits of n. We show that if n has at least 22 digits and satisfies the divisibility condition “n divides PD(n) minus 1”, then n must be a repunit, that is, all of its digits are equal to 1. The proof is purely a size comparison: for 22 or more digits, the maximum possible product of digits is already smaller than the smallest integer with that many digits, so the only way the divisibility can hold is when PD(n) equals 1. The remaining finite range of digit lengths from 1 to 21 can be checked by computer; we include a short Python script that performs an exhaustive search up to 6 digits (and can be extended further) and report that no non-repunit examples were found. We also briefly explain how the same argument works in any integer base greater than or equal to 3.
Author: Caraccioli Abrego, Ricardo Adonis
Institution: National Autonomous University of Honduras
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7105764055
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