A Pythagorean Introduction to Number Theory

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.

Ramin Takloo-Bighash is a Professor of Mathematics at the University of Illinois at Chicago, where his research is centered on Diophantine geometry and automorphic forms. His enthusiasm for research inspires his teaching at all levels.

"The book reads well. ... I would think this would be an enjoyable book from which to teach, since it covers the standard material in the way in which mathematics is done by asking questions and then developing the theory necessary for answering the questions. ... All in all, this is a very good antidote to the definition-theorem-proof approach to introductions to various subdisciplines of mathematics." (Duncan A. Buell, Mathematical Reviews, February, 2020)

"The present book has a high didactic quality being a detailed proof that number theory is 'a field of study that sits at the crossroads of many branches of mathematics, and that fact makes a prominent appearance in this book' as is pointed out in Preface. ... this book develops step by step a large number of techniques to solve a lot of number theoretic problems. ... this book is an excellent source for all readers interested in number theory." (Mircea Crâsmareanu, zbMath 1410.11002, 2019)"This book offers an interesting variation on the traditional undergraduate number theory course. ... The book is quite nicely written, with good motivation and a substantial supply of examples. ... the book has several other potential uses: it could be used as a text for a second semester course in number theory or 'special topics' course, or as a text for an introductory graduate course. It's also just an interesting book to have on one's shelf." (Mark Hunacek, MAA Reviews, June 24, 2019)