Homework 2

This assignment is due by the beginning of class on Wednesday, September 10^{th}. Be sure to review the homework guidelines before getting started.

- Prove that if $a^n-1$ is prime and $n>1$, then $a=2$.
- For an integer
*n*, let $\nu(n)$ be the number of positive divisors of*n*. For example, the positive divisors of 6 are 1, 2, 3, 6, and so $\nu(6) = 4$. If*n*factors as $n = p_1^{a_1}\cdots p_r^{a_r}$, determine $\nu(n)$ — with explanation. Is it true that $\nu(nm) = \nu(n)\nu(m)$ for any pair of integers*n*and*m*? - From Strayer, do the following problems:
- 19
- 30
- 43
- 44
- 54 (c,d)
- 55

It would also be good to get some practice on other problems, even though I won't be collecting them. For this, I'd suggest you take a look at the following problems

- Prove that if $a^n-1$ is prime, then
*n*is prime. Together with the first problem in this assignment, this means that $a^n-1$ only has a chance of being prime if $a=2$ and*n*is prime. - From Strayer, you might try
- 24
- 26
- 36
- 38
- 47

Any other problems you want to work on are also good practice, so if these problems all seem to easy, try out some others on your own.

page revision: 9, last edited: 29 Oct 2008 19:03