Homework 2

This assignment is due by the beginning of class on Wednesday, September 10th. 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.

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