Notes on Proofs:

Chapter 1: Divisibility, the Euclidean algorithm, primes and the Fundamental Theorem of Arithmetic, proofs of the infinitude of primes.

Chapter 2: Congruences, Euler's phi function, the ring of integers mod m, Fermat's little theorem and its generalization by Euler, Wilson's theorem, Fermat's theorem characterizing integers that are sums of two squares, polynomial equations mod m, the Chinese remainder theorem, RSA cryptography, Hensel's lemma, primitive roots, quadratic residues and higher residues.

Chapter 3: Quadratic reciprocity, the Jacobi symbol and applications, binary quadratic forms.

Chapter 4: Some functions of number theory, Moebius inversion formula.

Some elementary theorems on the distribution of prime numbers: (This part is not from the book) Chebyshev's functions, Abel's identity, Some equivalent forms of the prime number theorem, Shapiro's Tauberian theorem, Selberg's asymptotic formula.