\part{Advanced}



\chapter{Sum and product notation}


\[\sum^{5}_{i=1} 3\cdot i = 3 \cdot 1 + 3 \cdot 2 + 3 \cdot 3 + 3 \cdot4 + 3 \cdot 5 = 45\]


\[\prod^{5}_{i=1} 3\cdot i = (3 \cdot 1) \cdot (3 \cdot 2) \cdot (3 \cdot 3) \cdot (3 \cdot 4) \cdot (3 \cdot 5) = 262440\]

This notation is adapted for other operations in other areas of mathematics, like unions and intersections of set theory.

O





\chapter{Advanced algebraic identities}

\section{Binomial theorem}
 We begin with introducing theorems that facilitate expanding polynomials to some power; we first consider the expansion of $(x+y)^n$.

\begin{theorem}[Binomial theorem]
	\[ x,y \in \mathbb{R} \land n \in \mathbb{N} \implies (x+y)^{n}=\sum^{n}_{k=0} \binom{n}{k} x^{k} y^{n-k} \]
\end{theorem}

\section{Multinomial theorem}




Distributive property is what allows for many of the algebraic identities in part 3, as well as the binomial theorem and the identities we are to study now.

Many of the algebraic identities here are used as tools within the branch of elementary number theory.

We'll start by introducing an ancient identity known for almost 2 millenia.

\begin{theorem}[Brahmagupta-Fibonacci identity]
Let $a,b,c,d$ be real numbers, then
\[  (a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2=(ac+bd)^2+(ad-bc)^2 \]
\[a,b,c,d\in \mathbb{R} \implies  (a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2=(ac+bd)^2+(ad-bc)^2 \]
\end{theorem}

\begin{theorem}[Sophie Germain's identity]
Let $a,b,c,d$ be real numbers, then
\[x,y\in \mathbb{R} \implies x^4+4y^4=(x^2+2y^2+2xy)(x^2+2y^2-2xy) \]
\end{theorem}


\begin{theorem}[Euler's four-square identity]
Let $a_i,b_i$ be real numbers, then
\end{theorem}

\begin{theorem}[Lagrange's identity]
\end{theorem}

The following identity was devised to discuss properties of the Fibonacci sequence.
\begin{theorem}[Candido's identity]
	\[ [ x^2 + y^2 + (x+y)^2 ]^2 = 2[x^4 +y^4 (x+y)^4]\]
\end{theorem}