The Kaprekar Number, a class of natural numbers with unique structural properties in number theory, is characterized by its power results remaining equal to the original number when divided by equal-length fractions and summed. Starting with second-order Kaprekar numbers, this study extracts core patterns through number theory derivations and empirical validation. By extending these patterns to higher orders, we establish a cross-order verification framework. Focusing on fifth-order Kaprekar numbers, we conduct comprehensive analysis of both primitive and derived solutions, along with structural deconstruction. Key findings reveal: second-order Kaprekar numbers possess three core features—cyclic number origin, n-fold recursive patterns, and modulo-9 divisibility; their patterns maintain two essential attributes across orders: upgraded modulo operations and equal-length fraction properties; fifth-order Kaprekar numbers exhibit finite primitive solutions and infinite derived solutions, all decomposable into combinations of cyclic number bases and modified terms. The universality of these patterns is constrained by three factors: number base type, number field range, and order threshold. This paper details a rapid method for calculating repeated numbers of any cyclic number using second-order Kaprekar numbers (abbreviated as Kaprekar numbers), demonstrating how one pattern generates 60 fifth-order Kaprekar numbers through a single fifth-order Kaprekar number.

Zeng Junxiong. An Analysis of a Law That Achieves 60 Fifth-order Kaplerig Numbers—On the Second-order Kaplera Number. Academic Journal of Mathematical Sciences (2026), Vol. 7, Issue 1: 1-7. https://doi.org/10.25236/AJMS.2026.070101.

[1] TAN Xiangbai. Number: God's Pet [M]. Shanghai: Shanghai Education Press, 1996 (reprinted in 1999), pp. 83-87,169-172.

[2] Zeng Junxiong. The Wonderful Phenomenon of Cyclical Periods [J]. China Science and Education Innovation Guide, 2014(14):191-192.

[3] Zeng Junxiong. Addition Rules in Kapluga Numbers [J]. Curriculum and Education Research, December 2014 (late issue), 231-232.