Toward advanced number theory

Why is $13=2^2+3^2$, while 11 cannot be written as a sum of two integer squares? Reducing squares modulo 4 rules out 11 immediately: a square leaves remainder 0 or 1. Proving that every prime congruent to 1 modulo 4 is a sum of two squares requires a more substantial idea. This movement between a calculation, an obstruction, and a general existence theorem is characteristic of number theory. It also explains why elementary questions can lead toward algebra, analysis, or geometry without any one of those subjects supplying the whole story.

Begin with arithmetic

A good first spine is William Stein's Elementary Number Theory: Primes, Congruences, and Secrets. Begin with Chapters 1 and 2, especially the Chinese remainder theorem in Section 2.2. Solve a pair of congruences by hand, then explain why the solution is unique modulo the product when the moduli are coprime. Continue to Chapter 4 on quadratic reciprocity; Section 5.7 returns to sums of two squares. This gives a connected project rather than a collection of unrelated tricks. The computational examples are useful experiments, but the proofs explain which patterns persist.

You can start the arithmetic with induction, divisibility, and some experience writing proofs. Stein assumes some basic abstract algebra for the book as a whole, so keep quotient rings, finite groups, and fields available for review. To decide whether to skip the opening material, try proving Bézout's identity, finding an inverse modulo 35 when it exists, and explaining why cancellation modulo a composite number needs a hypothesis. If these are routine, move to reciprocity or continued fractions. If they are not, a short return to the Euclidean algorithm will be more useful than pushing through unfamiliar notation.

Three ways the subject opens up

The algebraic branch asks what happens when integers are replaced by algebraic integers. In $\mathbb Z[\sqrt{-5}]$, the two decompositions $6=2\cdot3=(1+\sqrt{-5})(1-\sqrt{-5})$ suggest that familiar factorization arguments need repair. James S. Milne's Algebraic Number Theory supplies a later route: use Chapter 1 to identify missing commutative algebra, then study Chapter 2 on rings of integers and Chapter 3 on Dedekind domains and ideal factorization. Before beginning, you should be able to use ideals, quotient rings, and finite field extensions in examples. Calculate in a quadratic field alongside the general theory; otherwise ideals can become names for a repair you have never seen working.

An analytic direction starts from different questions: how many primes lie below a large bound, or how evenly are they distributed among residue classes? Here estimates, series, and complex functions become central. Prepare real and complex analysis for this branch as those tools arise. Completing algebraic number theory first is unnecessary. Conversely, familiarity with congruences does not make contour integration or convergence arguments automatic. Choose a first analytic project small enough that you can distinguish an exact identity from an asymptotic statement and identify the error being controlled.

For a connection with geometry, Milne's Modular Functions and Modular Forms offers a later alternative. Read its introduction for motivation, then the preliminaries and the discussion of modular curves as Riemann surfaces before Section 4 on modular forms. Its stated preparation includes advanced undergraduate algebra and complex analysis. The boundary conditions at cusps matter just as much as the transformation formula; ignoring them changes the objects under study.

Keep examples within reach

Keep a notebook of worked examples and failed conjectures: noncoprime congruences, quadratic residues for small primes, and competing factorizations. For each theorem, record where its hypotheses enter. Then choose a next branch by the question you want to answer: continued fractions for approximation, ideals and valuations for arithmetic in fields, analytic estimates for distribution, or elliptic curves and modular forms for arithmetic geometry. There is no need to turn these branches into one compulsory reading sequence.

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