Cycles in Number Rows on the Example of the Set of Squares of Numbers and Their Practical Utility for Creating Code Locks

This work investigates cyclic patterns in the rows of numbers obtained from the sequence of squares n². While it is commonly known that the last digits of squares form a repeating cycle of length 10, this research goes further and examines the second, third, fourth, and fifth rows of digits from the end. Each of these rows is shown to form its own periodic cycle, with lengths increasing as 10, 50, 500, 5000, and 50000. All discovered cycles possess mirror symmetry with respect to two central digits. The theoretical justification is provided via modular arithmetic. The work also proposes an idea for applying these nested cyclic rows to create virtual code locks. Since the rows are repeating and nested within each other, they can be represented as closed rings that can be rotated and combined. The key to the code consists of the formula, the choice of rows, the shift of each ring, and the new order of numbers. A fake mechanism is also introduced to break the connection between numbers and shifts, making the system impassable for brute force. The study verifies the discovered pattern on several other formulas (cubes, fourth power, triangular numbers, linear equations), confirming that nested cycles always appear in the rows of numbers. The complete tables of Rows №1–5 are presented in a separate appendix available at: https://doi.org/10.5281/zenodo.23097688 Our model is not a replacement for cryptographic systems, but it can be used in tasks where deterministic code generation is required.