What question did this study set out to answer?

This work aims to generalize the Digit-Cycle Recurrence to n-digit scenarios and derive its maximum period properties.

February 6, 2026Open Access

Digit-Cycle Recurrence (DCR): n-Digit Generalization and Maximum Period Structure

Key Points

This work aims to generalize the Digit-Cycle Recurrence to n-digit scenarios and derive its maximum period properties.
Formalized DCR using linear congruence in modular arithmetic
Derived maximum period using matrix order principles
Applied the Chinese Remainder Theorem to support findings
Analyzed irregular cycles in relation to maximum period
Established a maximum period formula for n-digit DCR
Proved that all irregular cycles divide the maximum period
Extended previous fixed-digit studies to include arbitrary digit lengths and bases

Abstract

This paper presents an n-digit generalization of the Digit-Cycle Recurrence (DCR), a digit-based iterative transformation in base b. We formalize DCR as a linear congruence over (Z/bZ) ⁿ, derive the maximum period formula using matrix order and the Chinese Remainder Theorem, and prove that all irregular cycles divide the maximum period. This work extends previous fixed-digit studies and provides a unified theoretical framework applicable to arbitrary digit lengths and bases.

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