Homework 10

This assignment is due to my office by 11am on Wednesday, November 5th. 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
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