Through its engaging and unusual problems, this book demonstrates methods of reasoning necessary for learning number theory. Every technique is followed by problems (as well as detailed hints and solutions) that apply theorems immediately, so readers can solve a variety of abstract problems in a systematic, creative manner. New solutions often require the ingenious use of earlier mathematical concepts - not the memorization of formulas and facts. Questions also often permit experimental numeric validation or visual interpretation to encourage the combined use of deductive and intuitive thinking.
The first chapter starts with simple topics like even and odd numbers, divisibility, and prime numbers and helps the reader to solve quite complex, Olympiad-type problems right away. It also covers properties of the perfect, amicable, and figurate numbers and introduces congruence. The next chapter begins with the Euclidean algorithm, explores therepresentations of integer numbers in different bases, and examines continued fractions, quadratic irrationalities, and the Lagrange Theorem. The last section of Chapter Two is an exploration of different methods of proofs. The third chapter is dedicated to solving Diophantine linear and nonlinear equations and includes different methods of solving Fermat's (Pell's) equations. It also covers Fermat's factorization techniques and methods of solving challenging problems involving exponent and factorials. Chapter Four reviews the Pythagorean triple and quadruple and emphasizes their connection with geometry, trigonometry, algebraic geometry, and stereographic projection. A special case of Waring's problem as a representation of a number by the sum of the squares or cubes of other numbers is covered, as well as quadratic residuals, Legendre and Jacobi symbols, and interesting word problems related to the properties of numbers. Appendices provide a historic overview of number theory and its main developments from the ancient cultures in Greece, Babylon, and Egypt to the modern day.
Drawing from cases collected by an accomplished female mathematician,
Methods in Solving Number Theory Problems is designed as a self-study guide or supplementary textbook for a one-semester course in introductory number theory. It can also be used to prepare for mathematical Olympiads. Elementary algebra, arithmetic and some calculus knowledge are the only prerequisites. Number theory gives precise proofs and theorems of an irreproachable rigor and sharpens analytical thinking, which makes this book perfect for anyone looking to build their mathematical confidence.
這本書透過引人入勝且不尋常的問題，展示了學習數論所需的推理方法。每一種技術後面都跟隨著問題（以及詳細的提示和解答），這些問題立即應用定理，使讀者能夠以系統性和創造性的方式解決各種抽象問題。新的解法通常需要巧妙地運用早期的數學概念，而不是單純記憶公式和事實。問題也常常允許進行實驗性的數值驗證或視覺解釋，以鼓勵演繹思維和直覺思維的結合。
第一章從簡單的主題開始，如偶數和奇數、可除性和質數，並幫助讀者立即解決相當複雜的奧林匹克類型問題。它還涵蓋了完全數、友好數和圖形數的性質，並介紹了同餘。接下來的章節以歐幾里得算法開始，探討整數在不同進位制下的表示，並檢視連分數、二次無理數和拉格朗日定理。第二章的最後一部分探討了不同的證明方法。第三章專注於解決丟番圖線性和非線性方程，並包括解決費馬（佩爾）方程的不同方法。它還涵蓋了費馬的因式分解技術和解決涉及指數和階乘的挑戰性問題的方法。第四章回顧了畢氏三元組和四元組，並強調它們與幾何學、三角學、代數幾何和立體投影的聯繫。還涵蓋了瓦林問題的特殊情況，即用其他數的平方或立方和來表示一個數，以及二次剩餘、勒讓德符號和雅可比符號，以及與數的性質相關的有趣文字問題。附錄提供了數論的歷史概述，涵蓋了從古希臘、巴比倫和埃及的古代文化到現代的主要發展。
《數論問題解法方法》是由一位成就卓越的女性數學家收集的案例所編寫，旨在作為自學指南或一學期入門數論課程的補充教材。它也可以用來準備數學奧林匹克。唯一的先修知識是基礎代數、算術和一些微積分知識。數論提供了精確的證明和無可指責的嚴謹定理，並提升分析思維，使這本書非常適合任何希望建立數學自信的人。
Ellina Grigorieva, PhD, is Professor of Mathematics at Texas Women's University, Denton, TX, USA.
