A Pythagorean Introduction to Number Theory: Right Triangles, Sums of Squares, and Arithmetic

by Ramin Takloo-Bighash

Part I Foundational Material.- 1. Introduction.- 2. Basic number theory.- 3. Integral solutions to the Pythagorean Equation.- 4. What integers are areas of right triangles?.- 5. What numbers are the edges of a right triangle?.- 6. Primes of the form 4k+1.- 7. Gauss sums, Quadratic Reciprocity, and the Jacobi symbol.- Part II Advanced Topics.- 8. Counting Pythagorean triples modulo an integer.- 9. How many lattice points are there on a circle or a sphere?.- 10. What about geometry?.- 11. Another proof of the four squares theorem.- 12. Quadratic forms and sums of squares.- 13. How many Pythagorean triples are there?.- 14. How are rational points distributed, really?.- Part III Appendices.- A. Background.- B. Algebraic integers.- C. SageMath.- References.- Index.

300 pages, Paperback

Published July 10, 2019

Reviews

[Karl Lydon · Rating 2 out of 5]

After slogging through chapter 2 I am immediately met by a rant by the author on how "Euro-centric" the history of mathematics is made to be. This sticks out like a sore thumb and has essentially no reason to be in the text at all, the author seems to be using the notes as an excuse for going on his own personal vendetta. Maybe mathematics is seen from a European point of view because Europeans have provided the greatest contributions to mathematics? Something tells me the authors personal biases will not accept this to be the case.