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Exercises 9.6 Exercises

1.

Compute the group of units Un for n=10,11,12.

3.

Prove that if p is prime, then apa (mod p) for every integer a.

5.

Formally prove that ϕ(p)=p1 for prime p, by deciding which [a]{[0],[1],[2],,[p2],[p1]} have gcd(a,p)=1.

6.

Verify Euler's Theorem by hand for n=15 for all relevant a (note that ϕ(15)=8, and remember that a8=((a2)2)2 so we can use modulo reduction at each squaring).

7.

Get the inverse of 29 modulo 31, 33, and 34 using Euler's Theorem.

8.

Evaluate without a calculator 1149 (mod 21) and 139112 (mod 27).

9.

Solve the congruence 33x29 (mod 127) and (mod 128).

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 ϕ(N) where N=i=1kpiei. Then prove it by induction.

12.

Conjecture and prove a necessary (or even sufficient) criterion for when ϕ(n) is even. (Thanks to Jess Wild.)

13.

Compute the ϕ function evaluated at 1492, 1776, and 2001.

Let f(n)=ϕ(n)/n.

14.

Show that f(pk)=f(p) if p is prime.

15.

Find the smallest n such that f(n)<1/5.

16.

Find all n such that f(n)=1/2.

17.

Prove whether there are infinitely many values of ϕ that end in zero.

18.

Conjecture whether there are any relations between m and n that might lead ϕ(m) to divide ϕ(n).