Table of Contents: Chapter A. Divisibility, Unique Factorization Chapter B. Congruences, the CRT Chapter C. Primitive Roots Chapter D. Quadratic Residues Chapter E. Sums of Squares, and other Diophantine Problems Chapter F. Multiplicative Functions. Chapter G. Continued Fractions, Best Approximation. Chapter H. Pell's Equation Chapter J. Special Topics Chapter K. Z[i] and other quadratic rings Chapter L. Classical Factorization and Primality Tests. November 1 2009: Rewrote section on the Index Calculus algorithm. March 4 2009: Typographical changes. March 2 2009: Chapter L. Will review the whole book, and possibly adjust some bad page breaks. March 1 2009: Chapters J, K. A few corrections in CH. L, and some typographical details remain. Feb. 28 2009: Chapter H completed. Feb. 27 2009: Chapters G and H. Feb. 26 2009: Chapters A-F corrected. Feb. 25 2009: Today's work concentrated on Chapter L. Feb. 24 2009: Having received a long list of corrections I'm into another round of debugging, hopefully the last. This will take one or two weeks. Today I worked mainly on Chapters A and B. Sept. 20 2008: Ch. J, minor corrections. Sept. 19 2008: Ch. H, K, L corrected. Sept. 18 2008: Worked on Ch. E-G . Corrected minor mistakes. Sept. 17 2008: debugged Ch. A-D. Rewrote parts of "Zolotareff's Proof" in the interest of clarity. Fall of 2008 Having a period of further debugging. On Sept. 16 (hopefully!) completed most of Chapters A-C. Will continue at the rate of one or two Chapters each day. October 8, 2007, expanded Section on QS. September 19, 2007 Debugged Chapter L, added Section on QS. September 16,2007 The text has lain dormant for more than half a year. I recently took it up again and made several minor additions and corrections. By the end of the month I will finally try to find a publisher. Jan 6, 2007 Current state of the book: All text written, first five Chapters debugged by two very careful readers. Apart from corrections only small adjustments may remain. Haven't quite decided on the proper format for the tables at the end. Will probably try to find a publisher the coming Spring.