The Exponential Congruence Symbol
Es-said En-naoui
TL;DR
The paper introduces the Exponential Congruence Symbol $(a/n)_k$, a three-valued generalization of classical residue symbols that encodes when $a^k\equiv1$ or $-1\pmod{n}$. It develops a comprehensive algebraic framework, proving properties such as multiplicativity, power-compatibility, and inversion, and shows how the symbol decomposes via the Chinese Remainder Theorem and relates to multiplicative orders and subgroup structures. Connections to Legendre/Jacobi symbols, Dirichlet characters, and potential links to Dirichlet $L$-series are established, together with interpretations in cyclic groups and field extensions. The work provides a unified perspective on higher power residues and offers tools for congruence solvability, primality testing, and cryptographic applications, while outlining open problems and directions for further analytic and algebraic study.
Abstract
In this work, we study the generalized k-th power symbol (a/n)_k and present a comprehensive collection of its algebraic properties. The results are classified according to their dependence on the three main parameters a, n, and k. In particular, we discuss multiplicativity, inversion, power compatibility, and invariance modulo n with respect to the parameter a. For n, we examine factorization properties, behavior on prime powers, orthogonality relations, and Kummer splitting criteria. Regarding k, we include specialization to classical symbols, k-th reciprocity laws, relations between orders, and embedding into roots of unity. Moreover, we extend the existing theory by providing new essential results, including additive behavior under characters, Mobius filtering, compatibility with Carmichael and Euler functions, and connections with Dirichlet L-series. Finally, we analyze the case where a, n, and k are primes and present mixed results that generalize classical reciprocity laws, Frobenius automorphisms, and Sato-Tate distributions. These results unify and extend previous studies on k-th power symbols and offer a foundation for further arithmetic, algebraic, and analytic investigations.