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. Skryť popis