Exercises 17.7 Exercises
1.
Evaluate the Legendre symbols for
2.
Use the previous problem, your knowledge of
3.
Do any Legendre symbols in Example 17.1.5 which you didn't already do.
4.
Make up several hard-looking Legendre symbols
5.
Use the multiplicative property of the Legendre symbolβ3β to give a congruence condition for when
6.
For
7.
In Exercise 16.8.9, you explored
In Example 17.5.4 there are a number of small issues which need proof; here, you have the opportunity to finish them off.
8.
Let
9.
Prove: if
10.
Prove that for any prime
11.
Verify the previous exercise for
12.
Prove that if
13.
Verify Fact 17.4.10 by coming up with four Jacobi symbols which evaluate to
14.
Learn about the GoldwasserβMicali public key encryption method. How is it implemented? What mathematics from this chapter is used?
15.
Make up and compute some Legendre symbols that seem pretty hard by using the Jacobi symbol instead.
16.
If you didn't do them already, do the exercises in Example 17.4.8.
17.
Evaluate five non-obvious Legendre symbols
18.
Find congruence criteria for
19.
Use quadratic reciprocity to find a congruence criterion for when
20.
Use quadratic reciprocity to prove the surprising statement that
21.
Use Sage to explore why repetition in the decimal expansion of
22.
Explore the Solovay-Strassen primality test. Try implementing it well enough to check whether a number other than
23.
Compute two nontrivial (that is, not obviously perfect square) Jacobi symbols for the odd composite number