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February 6, 20260 citationsOpen Access

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

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SPSeon-Woo Park

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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Cite This Study

Seon-Woo Park (2026) studied this question.

synapsesocial.com/papers/698585fe8f7c464f23009df3https://doi.org/10.5281/zenodo.18478130
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