Solvability of Congruence Equations and Potential Applications in Physics

This paper investigates the explicit solvability of the congruence equation 10n≡1(mod9n). By exploiting properties of repdigit Niven numbers, we introduce several infinite families of primes with the square-free property and memory, thereby providing infinitely many explicit solutions to the congruence. Furthermore, we construct infinitely many new explicit Niven numbers. The results can be extended to the most general congruence equations an±bn≡0(mod(a±b)n) for arbitrary coprime integers a and b. The results reveal a recursive structure linking the equation to prime factorizations of repunits, and we propose several conjectures concerning the square-free property of the arising prime sets. Beyond pure number-theoretic value, the derived hierarchical prime families and modular periodic structures may have broad potential physical application including discrete quantum circuit design, quasi-crystal diffraction spectrum modeling, periodic photonic grating engineering and so on.