Exercises 9.6 Exercises
1.
Compute the group of units
2.
Prove Theorem 7.5.3 as a corollary of Theorem 9.2.5.
3.
Prove that if
4.
Use Exercise 9.6.3 to prove the polynomial
5.
Formally prove that
6.
Verify Euler's Theorem by hand for
7.
Get the inverse of 29 modulo 31, 33, and 34 using Euler's Theorem.
8.
Evaluate without a calculator
9.
Solve the congruence
10.
Solve as many of the systems of congruences we already did Exercises 5.6 using the Chinese Remainder Theorem and Euler's Theorem as you need in order to understand how it works. Follow the models closely if necessary.
11.
Use the facts from Section 9.5 to create a general formula for
12.
Conjecture and prove a necessary (or even sufficient) criterion for when
13.
Compute the
17.
Prove whether there are infinitely many values of
18.
Conjecture whether there are any relations between
19.
Look up the Carmichael conjecture about