8 New Number Theory Books Reshaping the Field in 2025

Ian Stewart and David Tall bring a fresh perspective to algebraic number theory by intertwining it with the fascinating saga of Fermat's Last Theorem. This book delves into the core concepts of algebraic numbers, guiding you through advanced topics like Galois theory, cyclotomic fields, and ramification theory, all updated to reflect the latest research. You'll explore how Wiles's proof not only solved a historic puzzle but also opened avenues for new mathematical inquiries. If you're eager to deepen your understanding of algebraic structures and their applications in number theory, this edition offers both foundational knowledge and insights into cutting-edge developments.

Unlike most number theory books that focus purely on theory, Peter Shiu's approach integrates computational techniques directly with foundational concepts, making this text particularly engaging for anyone wanting to see number theory in action. You’ll explore classic topics like divisibility, congruences, and cryptography alongside analytic results such as Dirichlet’s theorem and the prime number theorem, all reinforced with Python algorithm implementations. Shiu also ventures into less commonly covered areas like Aubry’s theorem and the Tonelli–Shanks algorithm, enriching your understanding beyond standard texts. This book suits undergraduates comfortable with basic calculus and coding who want to apply theory through computation, rather than those seeking a purely abstract treatment.

This tailored book explores the latest breakthroughs in number theory emerging in 2025, focusing on your expertise and interests. It covers cutting-edge developments across algebraic, analytic, and computational realms, revealing how contemporary research reshapes classical and modern perspectives. By matching your background and goals, the book dives into new discoveries, recent innovations, and evolving theories that define the current frontier in number theory. Through this personalized approach, you engage deeply with the most relevant topics—from modular arithmetic to transcendental number theory—enhancing your understanding and enabling you to stay ahead in this dynamic field. It offers a unique learning experience that aligns precisely with what you want to explore and achieve.

After extensive research in transcendental number theory, Vladimir G Chirskii developed this book to introduce the extension of the Siegel-Shidlovskii method to Euler-type series, a novel class of F-series. You’ll explore how arithmetic properties of infinite-dimensional vectors connect with p-adic integers and learn about Hermite-Padé approximations applied to hypergeometric series with algebraic and transcendental parameters. The book offers detailed frameworks to investigate the values of these series, presenting both classical theorems and fresh directions for inquiry. If you’re delving into advanced transcendental numbers or seeking new approaches to longstanding problems, this text provides a technical yet rewarding perspective.

Drawing from a multifaceted international collaboration, this volume captures the swift evolution of algebraic and analytic number theory. It dives into complex areas like elliptic curves in cryptography and Hopf Galois theory, offering you a panoramic view of cutting-edge research across seven countries. You’ll find chapters that don't just review established knowledge but push into fresh territories, making it a solid resource if you're involved in or curious about contemporary mathematical research. While highly technical, the book suits researchers and advanced students aiming to deepen their understanding of current trends and applications in number theory.

When Pio J Arias set out to write this book, he aimed to demystify the foundational elements of number theory by focusing on core concepts like divisibility, prime numbers, and modular arithmetic. You’ll explore Diophantine equations and number-theoretic functions through clear explanations that build your problem-solving skills step by step. This book suits anyone curious about the mathematical properties of numbers, whether you’re a student or a math enthusiast looking to deepen your understanding of the subject’s basics. It’s especially useful if you want a straightforward introduction that connects theory with practical exercises.

This personalized book dives into the evolving landscape of Number Theory, focusing on the most recent advances emerging in 2025. It explores new discoveries and research tailored to your specific interests and goals, allowing you to engage deeply with areas such as algebraic, analytic, and computational number theory. By concentrating on the latest trends, this tailored guide reveals how contemporary breakthroughs connect to foundational concepts, helping you stay at the forefront of this dynamic field. Whether your background is theoretical or applied, the content matches your experience and emphasizes developments that matter most to you.

When Humbert Cole explored the intricacies of modular arithmetic and gcd properties, he crafted a focused guide that distills fundamental proofs and algorithms in Number Theory. You’ll learn how the modulo operation works alongside modular multiplication and the binary exponentiation algorithm, with concrete examples illustrating these concepts. The book walks you through the Euclidean algorithm to find the greatest common divisor, then deepens your understanding by proving how gcd and lcm distribute over each other—concepts often glossed over elsewhere. If your interest lies in mastering the mathematical structure behind divisibility and linear combinations, this compact volume offers clarity without overwhelming detail.

Taha Sochi's diverse academic background in electronics engineering and physics shapes this introductory volume on number theory, designed with clarity and accessibility in mind. You’ll explore fundamental concepts and solved problems that require only a modest mathematical foundation, suitable for college and junior undergraduate students. The book structures material to support learners in both formal coursework and independent study, with chapters that carefully balance theory and practice, making abstract ideas tangible through worked examples. If you’re seeking to build a solid base in number theory without getting lost in overly technical jargon, this book aligns well with your goals, though those looking for advanced or research-level content might find it less suited.

Wadim Zudilin's deep expertise in analytic number theory shapes this book into a focused exploration of how complex analytic functions govern the properties of integers. You gain insight into classical and recent breakthroughs, such as the transcendence of π, Hilbert's seventh problem, and the irrationality of values related to the Riemann zeta function. The author balances rigorous theory with accessible explanations, making it suitable for graduate students or anyone self-studying advanced number theory. Chapters include exercises of varying difficulty and endnotes that guide you toward ongoing research areas, offering both foundational knowledge and pathways to current developments.