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Section9.6Exercises

2

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

3

Formally prove that \(\phi(p)=p-1\) for prime \(p\), by deciding which \([a]\in \{[0],[1],[2],\ldots,[p-2],[p-1]\}\) have \(\gcd(a,p)=1\).

4

Verify Euler's Theorem by hand for \(n=15\) (note that \(\phi(15)=8\), and remember that \(a^8=((a^2)^2)^2\) so we can use modulo reduction at each squaring).

5

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

6

Evaluate without a calculator \(11^{49}\) (mod \(15\)) and \(139^{112}\) (mod \(27\)).

7

Solve the congruence \(33x\equiv 29\) (mod \(127\)) and (mod \(128\)).

8

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.

9

Use the facts from Section 9.5 to create a general formula for \(\phi(N)\) where \(N=\prod_{i=1}^k p_i^{e_i}\). Then prove it by induction.

10

Compute the \(\phi\) function evaluated at 1492, 1776, and 2001.

Let \(f(n)=\phi(n)/n\).

11

Show that \(f(p^k)=f(p)\) if \(p\) is prime.

12

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

13

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

14

Prove whether there are infinitely many values of \(\phi\) that end in zero.

15

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

16

Look up the Carmichael conjecture about \(\phi\). What does it say, and what is the current status of this conjecture?

17

Use the ideas that proved \(\phi\) was multiplicative (Subsection 9.5.2) to see whether you can finally solve A First Problem. Especially think of making a table.