are of the
form 4k+1.
are of the form
8k+1 or 8k-1.
are of the form 12k+1 or 12k-1.
are of the form 20k+1, 20k-1,
20k+9 or 20k-9.
The elementary properties of quadratic congruences and a method for their solution were studied in a previous chapter. Now, we focus our attention on some deeper properties of numbers that were discovered by Euler, Legendre, and Gauss. The simplest of these are the following.
are of the
form 4k+1. are of the form
8k+1 or 8k-1.
are of the form 12k+1 or 12k-1. are of the form 20k+1, 20k-1,
20k+9 or 20k-9.
Generalizing these results, Euler conjectured that the prime divisors p
of
numbers of the form 
are of the form 
or 
, for some
odd b. This is the Quadratic Reciprocity Law. The first complete proof
of
this law was given by Gauss in 1796. Gauss gave eight different proofs of the
law and we discuss a proof that Gauss gave in 1808. The chapter concludes with a
discussion of the Jacobi symbol and its applications.