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There will be four books in this trilogy:

Preface: General Preface Why study number theory?. Why give proofs?. Motivation and expectations Homework. Further exploration. The cover. Gauss's Disquisitiones Arithmeticae The language of mathematics. 
Chapter 0: Preliminary Chapter on Induction. Appendix 0A: A closed formula for sums of powers Homework: Problems and Hints 
Chapter 1: The Euclidean Algorithm. Appendix 1A: Reformulating the Euclidean Algorithm Homework: Problems and Hints 
Chapter 2: Congruences Appendix 2A: Congruences in the language of groups Homework: Problems and Hints 
Chapter 3: The basic algebra of number theory Appendix 3A: Factoring binomial coefficients, and Pascal's Triangle mod p Homework: Problems and Hints 
Chapter 4: Multiplicative functions Appendix 4C: Irreducible polynomials mod p Homework: Problems and Hints 
Chapter 5: The Distribution of Prime Numbers Appendix 5A: Bertrand's Postulate and beyond Homework: Problems and Hints Conway's Prime producing machine (Details in section 5.20). Search for Ulam's spiral. 
Chapter 6: Diophantine problems Appendix 6A: Polynomial solutions of Diophantine Equations Homework: Problems and Hints 
Chapter 7: Power Residues Appendix 7A: Card Shuffling and Fermat's Little Theorem Homework: Problems and Hints 
Chapter 8: Quadratic residues Appendix 8A: Eisenstein's proof of quadratic reciprocity Homework: Problems and Hints 
Chapter 9: Quadratic equations Appendix 9D: Descent and the quadratics Homework: Problems and Hints 
Chapter 10: Square Roots and Factoring Appendix 10A: Pseudoprime tests using square roots of 1 Homework: Problems and Hints 
Chapter 11: Rational approximations to real numbers Appendix 11A: Uniform distribution Homework: Problems and Hints 
Chapter 12: Binary quadratic forms Appendix 12A: Composition rules: Gauss, Dirichlet and Bhargava Homework: Problems and Hints 
Chapter 13: The anatomy of integers Appendix 13A: Other anatomies Homework: Problems and Hints 
Chapter 14: Counting integral and rational points on curves, mod p Appendix 14A: Gauss sums Homework: Problems and Hints 
Chapter 15: Combinatorial number theory Appendix 15A: Summing sets (mod p) Homework: Problems and Hints 
Chapter 16: The padic numbers Chapter 16.7: The padic dilogarithm Homework: Problems and Hints 
Chapter 17: Rational points on elliptic curves Appendix 17A: General Mordell's Theorem Homework: Problems and Hints 
References: The Great Books of number theory. Recommended Further Reading 