What question did this study set out to answer?

The aim is to explore an exact arithmetic invariant connecting the Kaprekar constants 495 and 6174.

March 17, 2026Open Access

An Exact Arithmetic Invariant Linking the Kaprekar Constants 495 and 6174

Key Points

The aim is to explore an exact arithmetic invariant connecting the Kaprekar constants 495 and 6174.
Analysis through factorization and ratio methods
Investigating structural relationships
Examining fixed points of Kaprekar routines
Reveals an exact invariant linking the two Kaprekar constants
Establishes a structural relationship independent of classical methods
Provides new insights into the nature of Kaprekar constants

Abstract

This note presents an exact arithmetic invariant linking the Kaprekar constants 495 and 6174 through factorization and ratio analysis. The result reveals a rigid structural relation between the unique fixed points of the 3-digit and 4-digit Kaprekar routines, independent of the classical iterative process. To the best of the author's knowledge, this invariant relation does not appear in the existing literature. The work highlights a direct arithmetic correspondence that may contribute to the structural understanding of Kaprekar constants in recreational number theory.

An Exact Arithmetic Invariant Linking the Kaprekar Constants 495 and 6174

Key Points

Abstract

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