Math Papers: Number Theory & Representation Theory - Nov 2025

Hey math enthusiasts! Are you looking to stay on top of the latest research in number theory and representation theory? You've come to the right place! This article dives into the newest papers from ArXiv's math.NT and math.RT categories, published as of November 3, 2025. We'll be breaking down the abstracts and highlighting key themes so you can quickly grasp the cutting-edge advancements in these fascinating fields. Don't forget to check out the Github page for even more papers and detailed information. Let's get started!

Latest Papers in Number Theory (math.NT) and Representation Theory (math.RT)

Wilson's Theorem Modulo Higher Prime Powers III: The Cases Modulo $p^6$ and $p^7$ (2510.26743)

In this intriguing paper by Bernd C. Kellner, the focus is on extending previous work related to Wilson's theorem. This time, the exploration delves into the complexities of Wilson's quotient modulo $p^5$ and $p^6$, which directly translates to understanding $(p-1)!$ modulo $p^6$ and $p^7$, respectively. Additionally, the research identifies certain power sums of Fermat quotients up to modulo $p^6$. What makes this study particularly compelling is the examination of patterns within the $p$-adic coefficients of the Wilson quotient and $(p-1)!$, showcasing how the fundamental congruence $(p-1)! ext{≡} -1 ext{ (mod } p)$ seamlessly integrates into the broader theoretical framework. This research builds upon earlier investigations into Wilson's theorem, pushing the boundaries of our understanding of number theory. The abstract highlights the computational aspect of determining the Wilson quotient and factorials modulo higher prime powers, emphasizing the intricate patterns that emerge in the $p$-adic coefficients. This meticulous analysis provides valuable insights into the behavior of these fundamental number-theoretic objects. For those interested in the finer details of number theory, this paper offers a wealth of information and a deep dive into the intricacies of Wilson's theorem. Kellner's work not only expands our computational capabilities but also sheds light on the theoretical underpinnings of these congruences, making it a significant contribution to the field. The identification of patterns in the $p$-adic coefficients opens up avenues for further research and potential generalizations, highlighting the lasting impact of this study. The paper's comprehensive approach, combining computational results with theoretical analysis, makes it a valuable resource for researchers in number theory and related fields.

Wilson's Theorem Modulo Higher Prime Powers II: Bernoulli Numbers and Polynomials (2509.05402)

Bernd C. Kellner continues his deep dive into Wilson's theorem in this paper, linking it with Bernoulli numbers and polynomials. The core idea revolves around expressing Wilson's theorem and the Wilson quotient using supercongruences of power sums of Fermat quotients modulo higher prime powers. These congruences are then translated into congruences involving power sums and Bernoulli numbers, leading to concise proofs compared to previous methods. As a practical application, the paper demonstrates the computation of the Wilson quotient up to modulo $p^4$ and the factorial $(p-1)!$ up to modulo $p^5$. This method can be extended to higher prime powers with further effort. A notable byproduct is the determination of power sums of Fermat quotients up to modulo $p^4$. This research showcases the power of connecting different mathematical concepts to gain deeper insights. The use of Bernoulli numbers and polynomials provides a new lens through which to view Wilson's theorem, revealing unexpected connections and simplifying complex calculations. The computation of the Wilson quotient and factorial modulo specific prime powers serves as a concrete example of the theory's practical applications, making the paper highly relevant for researchers and students alike. Kellner's work provides a clear and accessible pathway for understanding the interplay between Wilson's theorem, Bernoulli numbers, and Fermat quotients. The congruences derived in this paper offer a powerful tool for exploring number-theoretic relationships and solving problems in the field. The paper's structure, with its clear presentation of theoretical results and concrete examples, makes it a valuable resource for anyone interested in number theory. The extension of these methods to higher prime powers suggests further research avenues and the potential for even deeper understanding of these mathematical concepts.

Distribution of Local Signs of Modular Forms and Murmurations of Fourier Coefficients (2409.02338)

Kimball Martin's paper delves into the fascinating world of modular forms, exploring the distribution of local signs and the intriguing phenomenon of