Volume 3: Number Theory (Rigorous Elementary Mathematics)

In mathematical contests and olympiads, there are four subject areas: algebra, counting, number theory, and geometry. These areas are addressed in the four books of the Rigorous Elementary Mathematics series. While there are many books that are geared towards helping students to learn how to solve problems on contests, those books tend to assume prerequisite knowledge without proofs, and they instead focus on applying theorems to problems. This 4-volume series fills the gap in the literature by proving many of the relevant theorems in a logically sequenced framework from the ground up. Throughout this series, three ideas are reinforced: writing the same object in multiple ways, breaking up equality using antisymmetry, and using equivalence relations. It is suggested that the books be read by those who have experience with mathematical proofs and problem-solving. The ideal audience consists of teachers of mathematics who want to solidify their own fundamentals, and outstanding students who want a second look at the material. Volume 3: Number Theory begins with divisibility, prime numbers, and arithmetic functions. Most of the volume is about modular arithmetic, with chapters on the classic theorems of Euler, Fermat, and Wilson, modular exponentiation, and modular power residues. There are also chapters that pay special attention to base representations and Diophantine equations. Finally, these methods are applied to the study of combinatorial expressions, such as the theorems of Legendre, Kummer, and Lucas, and integers in particular forms, such as Fermat, Mersenne, and perfect numbers. Complete proofs are provided in the last chapter for the lifting the exponent lemma and Zsigmondy’s theorem via cyclotomic polynomials.