Bonus for Test 2

The following problem can be turned in for bonus on the second midterm. The first was originally a homework problem, but a typo in the original assignment might have left lots of people confused. You can turn this in before 11am on Friday to be graded as a bonus problem. As before, I won't be giving partial credit: only giving full credit or no credit at all. You should complete these problems entirely on your own: do not consult your friends or internet sources. If it seems to me that you've consulted outside sources on any one problem, you will receive no credit on your entire submission. You are, however, allowed to consult a calculator, the instructor, or your notes and text.

• 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 $\mod{p}$?)
• Determine whether or not $F_5 = 2^{2^5}+1$ is prime. Make sure to give a complete justification.
• Find a Mersenne number $M_n = 2^n-1$ with $n>50$ such that $M_n$ is composite. The larger your choice of n, the better. Make sure you give a complete justification.
• Prove that $F_n = 2^{2^n}+1$ is congruent to 17 mod 240 provided $n \geq 2$.