- Turn in February 11th: Write down both the
**converse**and**contrapositive**of the statement:If \(x^2 = x + 1\), then \(x = 1 + \sqrt{5}\) or \(x = 1 - \sqrt{5}\)

- Turn in February 13th:
Prove the following: If \(x^3\) is irrational then \(x\) is irrational.

- Turn in February 16th:
Let \(n = ab\) be the product of positive integers \(a\) and \(b\). Prove that either \(a \leq \sqrt{n}\) or \(b \leq \sqrt{n}\).

- Turn in February 18th:
Show that \( (p \to q) \to r\) is not logically equivalent to \( p \to (q \to r)\)

- Turn in February 20th:
Simplify the following if possible: \( (p \to (q \to r)) \to ((p \to q) \to (p \to r))\)

- Turn in February 23rd:
Let \(a\) be an integer. Prove that there exists an integer \(k\) such that \(a^2 = 3k\) or \(a^2 = 3k + 1\).

- Turn in February 25th:
Let \(x,a\) be integers with \(a \geq 2 \) such that \(a \mid 11x + 3\) and \(a \mid 55x + 52\). Find \(a\).

- Turn in February 27th:
Tell me how many natural numbers \( 1 \leq k \leq 14\) smaller than 15 have the following property \( \gcd{(k, 15)} = 1 \). Just for your information, we say two numbers \(a, b\) are

**relatively prime**when \( \gcd{(a,b)} = 1\). Also (for your information only) the number of relatively prime natural numbers smaller than \(a\) is called \( \phi(a) \) the**Euler Totient Function**. So this question is asking you to calculate \(\phi(15)\). - Turn in March 2nd:
Determine whether 9833 is prime.

- Turn in March 4th:
Show that for every natural number \(k\), \(6^k\) will never end in a 0 (using a decimal representation).

- Turn in March 13th:
Prove that \( 6 | n^3 + 5n\) for all \( n \geq 1 \)

- Turn in March 16th:
Find a (closed-form) formula for the following sum: $$ (0 - 0) + (1 - 1) + (8-4) + (27-9) + \cdots + (n^3 - n^2) $$ Use induction to prove your result.

- Turn in March 18th:
Find this sum: $$ S = -56 - 49 - 42 - \cdots + 623 + 630 + 637 $$

- Turn in March 20th:
Which, if any, of the following three sets are equal? $$ A = \{ k \mid k \in \mathbb{N}, 3 \nmid k, k \leq 12 \} $$ $$ B = \{ 3k + 1 \mid k \in \mathbb{N}, k \leq 4 \} $$ $$ C = \{ k \mid k \in \mathbb{N}, \gcd{(k, 3)}=1, k \leq 11 \} $$

- No Problem for March 23rd
- Turn in March 25th
Given the set \(A = \{1, 2, 3\}\) create a binary relation on \(A\) (that is, a subset of \(A \times A\)) which is

**reflexive**,**symmetric**, and is**not transitive** - Turn in March 27th
Prove that \(\overline{a} + \overline{b} = \overline{a+b}\) where \(\overline{a}\) is the equivalence class of \(a\) under the relation on \(\mathbb{Z}\) given by \(x \sim y\) when \(p | (x-y)\). Here \(\overline{a} + \overline{b}\) means the set \(\{ x + y \mid x \sim a \land y \sim b \}\), made by summing anything in \(\overline{a}\) and anything in \(\overline{b}\).

**This is an elaborate way of asking if computing \( (a + b) \textrm{ mod } p = (a \textrm{ mod } p) + (b \textrm{ mod } p) \).** - Turn in April 6th
None, happy spring break.

- Turn in April 8th
Let \(S = \{ 1,2,3,4\}\) and \(f,g : S \to S\) by \(f = \{(1,3),(2,2),(3,4),(4,1)\}\) and \(g = \{(1,4), (2,3), (3,1), (4,2)\}\). Find \(g \circ f \circ g^{-1} \).

- Turn in April 13th
Find \(x\) such that \(79x \equiv 15 \mod{ 722 }\)

- Turn in April 15th
Find \(6^{128} \mod{ 13 }\)

- None on April 17th
- Turn in April 20th
Problem 5 from math.prof.ninja/417, the in-class problems from Friday.

- Turn in April 22th
Problem 8 from math.prof.ninja/420, the in-class problems from Monday.

- Turn in April 24th
Problem 6 from math.prof.ninja/422

- Turn in April 27th
The last problem from 424

- Turn in April 29th
Give a method for detecting if a graph is bipartite.

- Turn in May 1st
How many 6 digit integers have 1 digit repeated 3 times, another distinct digit repeated twice, and a third digit? For instance 292129 or 877666.

- Turn in May 4th
Prove that the Petersen graph is not Hamiltonian

- Turn in May 6th
Create a graph in which any walk on the graph of length \(k\) is a length \(k\) word made from 'a' and 'b' with at most two 'b's in a row. Use this graph to decide the number of words of length 4 with this property.

- May 8th Test 3, good luck.
- May 11th, new topic no daily challenge.
- May 13th:
Problem 4 from Monday's problem set

- May 15th:
What is the worst case complexity of insertion sort?

- May 18th, final class:
Make a list of the super-powers you developed (or are developing) in this class.