Homework 10

This assignment is due to my office by 11am on Wednesday, November 5^th. Be sure to review the homework guidelines before getting started.

• Using only Lagrange's Theorem and Fermat's Little Theorem, prove Wilson's Theorem. (Hint: Consider the polynomial $f(x) = (x^{p-1}-1)-\prod_{i=1}^{p-1}(x-i).$ What is its degree? How many roots does it have $\mod{p}$?) (Moved to bonus)
• Determine whether $2^{2^3}+1}$ is a prime number using the primality test for Fermat numbers we derived in class.
• Verify that the conditions of our primality test fail when $n=6263$ and $a=2$. Does this mean that $6263$ is composite?
• From Chapter 5 of Strayer, do the following problems
  • 29(b)
  • 30(b,e)
  • 31(b)
  • 32(c)
  • 33(c)
  • 34
  • 36