Number theory has a seductive surface: its questions about primes and integers can be stated to a child, yet their answers can require the deepest machinery in mathematics. That gap is exactly why reading order matters. Reach for algebraic or analytic methods too soon and you drown; linger only on the elementary and you never see why they were invented.
The path that works is to master the elementary theory first, then branch into the algebraic and analytic approaches that modern number theory relies on, and finally savor the elegance that draws people to the subject. Each book below is placed to keep the difficulty rising at a manageable rate.
Master the elementary theory
Start with An introduction to the theory of numbers by Niven, Zuckerman, and Montgomery, a classic that covers divisibility, congruences, and quadratic reciprocity with clarity. Pair it with Number theory by George Andrews, a compact and elegant introduction that emphasizes partitions and generating functions. Together they give you the elementary toolkit that every later book assumes fluency in.
Branch into algebraic and analytic methods
Modern number theory splits into two great streams. On the algebraic side, A Classical Introduction to Modern Number Theory by Ireland and Rosen bridges elementary results to the algebraic viewpoint, and Algebraic number theory by Stewart and Tall is the accessible entry into number fields and ideals. On the analytic side, The distribution of prime numbers by Ingham is a concise classic on the prime number theorem and the analytic tools behind it. For a geometric flavor, An introduction to the geometry of numbers by Cassels shows how lattices illuminate number-theoretic problems, and Solving the Pell Equation explores the rich theory behind a single famous equation.
Reach depth and elegance
To see the field's craft at its finest, A Course in Arithmetic by Jean-Pierre Serre is a dense, beautiful book that packs modular forms and quadratic forms into a slim volume. And Proofs from the book by Aigner and Ziegler collects the most elegant proofs in mathematics, many number-theoretic, as a reminder of why the subject enchants.
Read in this order and number theory stops feeling like a set of tricks. Follow the full path to go from elementary congruences to the algebraic and analytic heart of the subject.
