Chapter 11
1. If p 5 (mod 8), then p = 8k + 5 and (p 1)/2 = 4k + 2. Note that
24k+2 (4k + 2)! = 2 4 6 (8k + 4)
(4k + 2)!(1)2k+1
(mod p)
Hence 24k+2 1 (mod p), so that by Eulers Criterion, 2 is a quadratic
nonresidue modulo p. The case p 7 (mod 8) is simi
ummryX rimes of peil porms
We consider numbers of three special forms, and ask the following three questions
of each:
1. Under what conditions is an integer of the given form always prime?
2. Under what conditions is an integer of this form always composi
ummryX ghinese eminder heorem
The focal point of this chapter is to nd and prove the Chinese Remainder Theorem, rst for two congruences, and then for n congruences. Along the way, we also
determine the exact conditions required for the pair of congruences