**Review**
The topic of congruences, first explored by Gauss, has two notable features. One is the geometric representation as integral points on a circle, with the number of points corresponding to the modulus (i.e. base). The second is some interesting theorems when one considers a prime number for the modulus.
Computation in any base is fairly straight forward if one first writes out the appropriate addition and multiplication tables to make sure that one is using the symbols properly.
The Fundamental Theorem of Arithmetic which states that every positive integer can be factored into a unique product of prime integers is both a "neat" theorem as well as a very powerful one, leading to many additional properties.
Another important theorem is known as The Prime Number Theorem. It was originally a conjecture by Gauss (1793?) and was finally proven in 1896 by two French mathematicians although Riemann (1859) is credited with mapping out the basic strategy of the proof. The Prime Number Theorem states that the distribution of prime numbers (i.e. the ratio p(n)/n ) is approximately 1/ln(n). |