Questions

1. Find all (complex) solutions to the equation \( 6z^5 + 15z^4 + 20z^3 + 15z^2 + 6z + 1= 0 \).
2. Consider the function \( f : \mathcal{P}(\mathbb{N}) \to \mathbb{N} \) defined by:

let \( p_n \) be the nth prime number, and let \( S = \{n_1, n_2, n_3, ...\} \) then \( f \) is given by:

\( f(S) = p_1^{n_1} * p_2^{n_2} * ... \)

We claim this implies there is a injection from the powerset of naturals to the naturals and so we provide a counterexample to Cantor's Theorem. Where is the mistake?

3. Find a closed form for \( \sum^n_{k=1} k \cos(kx) \).
4. Prove or give a counterexample: if a function \( f: \mathbb{R} \to \mathbb{R} \) is infinitely differentiable, and is nonzero somewhere on its domain, then it has countably many roots.
5. Show \( n! \) is not elementary.

1. Show that the Fundamental Theorem of Algebra and the statement that a polynomial of degree \( n \) has exactly \( n \) complex roots (including multiplicity), are equivalent.
2. Show that if \( G \) and \( H \) are groups of cardinality \( p \) where \( p \) is prime, then \( G \) and \( H \) are isomorphic.
3. Show that if \( R \) is a finite integral domain, then it is also a field (Wedderburn's little theorem).
4. Given any prime \( p \), show that \( \mathbb{F}_{p} \) is not algebraically closed.
5. Do the 32 bit integers under two's complement form a group under addition? If not what structure do they form?
6. Show the algebraic numbers form a subring of \( \mathbb{C} \). Do they form a field?

1. Show that a matrix is singular iff it has 0 as an eigenvalue.
2. Show that if \( M \) is an invertible matrix in some field \( K \), and has entries all in some subfield \( J \subseteq K \), then \( M^{-1} \) also has all of its entries in \( J \). What if \( J \) is not a field?
3. Let \( R \) denote the set of rows of a matrix \( A \), and \( C \) the set of columns. Prove or find a counterexample to: if \( R = C \) then \( A \) is symmetric.

1. Show that there are no solutions to the diophantine equation \( a^2 + b^2 = c^2 \) if both \( a \) and \( b \) are odd.
2. Show that \( \liminf\limits_{n \to \infty} \frac{\phi(n)}{n} = 0 \)
3. I choose a positive integer \( n \le 10000 \), your task is to determine whether or not this number is prime by "asking" a number \( k \), to which I will say "yes" if \( k \) is a divisor of \( n \), or "no" if it is not. What is the fewest number of "asks" needed so that you are guarenteed to know whether or not \( n \) is prime?
4. The powers of two \( \{ 2^0, 2^1, 2^2... \} \) are a very special sequence, any positive integer can be written as a sum of unique powers of two in exactly one way. Prove the powers of two are the only strictly increasing positive integer sequence with this property.

1. Show that \( \chi(C_{2n + 1}) = 3 \ \forall n \in \mathbb{Z}^+ \).
2. How many trees are there of size \( n \) up to isomorphism?

1. Find the smallest integer \( n \) such that \( n\ln{n} + n > 3^{197} \)
2. How many palindromes are there beneath \( 10^{100} \)? What about \( 100! \) ?
3. Let \( P_n = \{ s \in "([|])\{2n\}" : \text{s is parenthetically closed} \} \) show that \( P = \cup_{n \in \mathbb{N}} P_n \) is not a regular language.
4. Describe a list implementation (\( \text{append}(e: E): \text{unit}, \ \text{get}(idx): E, \ \text{remove}(idx): \text{unit} \)) with \( O(1) \) appends and removals.
5. What are the differences (if any) between the return values of these two functions:

def append1(x: List, y: List):
    x = x + [y]
    return x

def append2(x: List, y: List):
    x.append(y)
    return x