The Euler phi-function question?

Question

This function produces the count of positive integers less than or equal to n that are relatively prime to n . You are to make a table that keeps track of the first 50 values of n and then graph this function. Make sure to include 2 conjectures from your findings.

Answer 1

I can't graph the function here, but I can start a table
(I'll do half of it and let you do the other half).
n..........φ(n)
1...........1
2...........1
3............2
4............2
5............4
6............2
7............6
8............4
9............6
10...........4
11...........10
12............4
13............12
14.............6
15.............8
16.............8
17.............16
18..............6
19..............18
20...............8
21...............12
22................10
23.................22
24..................8
25..................20
You can carry on from here.
Here are my 2 conjectures about
Euler's function:
1). If φ(n) = k for some n, then there is at least
1 other value, n', such that φ(n') = k.
2. The only n such that φ(n) divides n-1 are
primes.
Both these conjectures are open. The first
is called Carmichael's conjecture and the
second Lehmer's conjecture.
In 1907 R.D. Carmichael thought he had
proved the first conjecture and even
gave it as an exercise in his book on
number theory. But his proof was faulty
and the problem is still open today.

Answer 2

That's much easier than it sounds. Just write down the numbers from 1 to 50, then count up all the numbers that don't have a common divisor. Include the number 1 as one of the mutally prime numbers.

For example, for the number 10, exclude 2, 4, 5, 6, and 8, leaving you with 1, 3, 7, and 9. So, φ(10) = 4.

As for the conjecture part, just make a couple of observations once you have your graph.

http://primes.utm.edu/glossary/page.php?sort=EulersPhi

Answer 3

McCullough