# Exam 3

Mind Map by jwyatt06, updated more than 1 year ago
 Created by jwyatt06 almost 5 years ago
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### Description

Sections 8.1, 8.2, 11.2, 11.3, 11.4, 12.1, and 14.2.

## Resource summary

Exam 3
1 Definition
1.1 order of an integer modulo p
1.2 Perfect Numbers
1.3 Mersenne primes
1.4 Fermat primes
1.5 Fibonacci sequence
1.6 Pythagorean triples, and primitive Pythagorean triples.
2 Prove
2.1 Theorem 8.1
2.2 The Lemma on pg. 221 regarding when a^k -1 is prime.
2.3 Theorem 14.2; will be given the identity (a) on page 289
2.4 Converse of Theorem 12.1 (top of page 249)
3 Understand
3.1 The cyclic nature of the list a, a^2, a^3, ... as well as the relationship between the repeats on the list and the order of a modulo p
4 Know
4.1 Theorem 8.3: The statement and the idea of the proof of the formula o(a^i)
4.2 When primitive roots exist, how many are there?
4.3 The idea in the proof of Lagrange's Theorem. In particular, if ab=0 (mod p), b=0 (mod p).
4.4 The statement of Theorem 8.6 (whose corollary gives the existence of primitive roots for primes.)
4.5 The statement of Theorem 11.1 regarding the existence of perfect numbers. (Prove the 'if' direction.)
4.6 Statement of Theorem 14.3
5 Be able to
5.1 Given an element of a certain order (mod p), use thm. 8.3 to produce elements of prescribed orders (example 8.1)
5.2 Problems in homework regarding perfect numbers.

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