Key Sequence

New Definitions

• division + division algorithm
• greatest common divisor
• prime
• Euclidean Algorithm

Results and Their Proofs

• principle of induction
• primes
  • there are infinitely many primes
• division and greatest common divisor
  • division algorithm
  • properties of the gcd
• Euclidean Algorithm and some euclid lemma linked below in fundamental theorem of arithmetic

Questions for Jana

• Why does something being prime require that \(p>1\). that is, why is \(1\) not defined as prime?

Interesting Factoids

\begin{equation} \mathbb{N} = \{0, 1,2,3 \dots \} \end{equation}

the set of natural numbers. start from 0.

\begin{equation} \mathbb{Z} = \{\dots, -2, -1, 0,1,2, \dots \} \end{equation}

the set of integers. natural language and their negatives

Key Sequence

• first, we built the ground work of principle of induction in order to construct the WOP
• we defined division, and formalized the algorithm for doing so
• we then defined the greatest common divisor, and the fact that greatest common divisor is a linear combination
• we then constructed the idea of prime numbers, coprimes, and showed that There are infinitely many primes
• Finally, we used yet another lemma from Euler to build the fundamental theorem of arithmetic

New Definitions

• division
  • division algorithm
  • greatest common divisor
• prime
  • coprime

Results and Their Proofs

• principle of induction
  • well-ordering principle
• There are infinitely many primes
• division algorithm
• greatest common divisor is a linear combination
• fundamental theorem of arithmetic
  • and its factorization motivator lemma

Key Sequence

New Definitions

• modular arithmetic + basic modular arithmetic operations
• permutation
• groups
  • left cancellation, right cancellation

Results and Their Proofs

• Chinese Remainder Theorem

Questions for Jana

• why is it that adding all the digits work for \(\ \text{mod}\ 9\). I still don’t get it.
• in re Chinese Remainder Theorem: is there any case in which \(a\ \text{mod}\ b\) is not unique?

Interesting Factoids

2nd order linear inhomogeneous: non-homogeneous linear differential equation

Sensitivity to Initial Conditions + Parameters.

ODE Existence and Uniqueness

We can recast all high order systems into a first-order vector-valued system. So, for any system:

\begin{equation} x’ = g(t,x, a) \end{equation}

if \(g\) is differentiable across \(t,x\) and \(a\), the IVP given by \(x’ = g(t,x,a)\) and \(x(0) = x_0\), has the property has that:

1. the ODE has a solution \(x(t_0) = x_0\) for any \(t_0\), and any two solutions on the interval coincide as the same solution
2. The only way for a solution to fail to extend temporally is due to the bounds’ \(||x(t)||\) becomes unbounded as \(t\) approaches the endpoints
3. On any interval \(t_0 \leq t \leq T\) the solution \(y_{a,y_0}\) depends continuously on \(a, y_0\), “if I look at my solution sometime later, it would be a non-discontinuous change on the choice of initial condition”

Example

Let’s consider: