Assignment 11

The most difficult part of assignment 11 was definitely the Jacobi and Legendre symbols, at first it seemed as though the Legendre symbol was just a continuation of the square-rooting section in modular arithmetic, but it soon developed into a new theorem without a clear-cut application apparent. The Jacobi symbols were more of a continuation of the Legendre symbol and were actually important because they related to the law of quadratic reciprocity which not only related to section 3.9 in roots of numbers but also might develop cryptographic apllications (or at least we hope so given its place in the book).

The most significant gem from the reading was actually not the material itself but a simple appreciate of the Jacobi symbols as a more complex form of the Legendre symbols much in the same vein as Euler's Theorem was a generalization of Fermat's theorem. It's common place in mathematics that the "easy" specific cases are often more generalized for universal application and it makes you wonder whether or not some "specific" cases for factoring numbers might soon generate a universal factoring algorithm that may make systems such as RSA obsolete.