In graph theory, directed graphs can be used to understand tournaments and theorems such as the king chicken theorem. First to understand the king chicken theorem, we will go over some terminology. A tournament is a directed graph that contains edges that have specific orientation. Tournament graphs are also used to show relationships between players and who beat who in a tournament. Complete graphs most often show this by using arrows. Any edge that points from \(i\) to \(j\) has directed orientation.

Some of the most important matrices that are used in number theory are known as the adjacency matrix and the transition matrix. An adjacency matrix is given by the vertices of that matrix and is labeled with a \(0\) or \(1\) depending on its adjacency. The way we label such a vertex with its adjacency is by \(\left(i,j\right)\), where \(i\) is the row while \(j\) is the column. Adjacency matrices can also be used to find the number of walks between vertices. To show this we raise our matrix to the \(L\), where \(L\) is the length of the walk and read off the matrix as \(\left(i,j\right)\).

RSA encryption can be used for many things such as keeping important messages secured. It is very difficult to break or decode messages that have been encrypted by RSA encryption if not given a public key. There are a few steps that one must go through in order to encrypt and decrypt a message.

We can look at few variables that are needed through the RSA encryption process:

\(e\) will be our public key

\(d\) is the value used for decoding and is only given to the receiver

\(p\) and \(q\) are the primes

\(n\) is the result of \(pq\)

\(M\) is the original message

\(C=M^e\) (mod \(n\)) is used to encrypt messages

\(C^d\) (mod \(n\)) is used to decrypt messages

In
discrete mathematics, graphs are used to show concepts of networks or
structures in a mathematical way. In particular, a graph consists of vertices
\(\left(V\right)\), that is a finite set and a set of edges \(\left(E\right)\). Each edge has at least one vertex
connected to it and can also be described as its endpoint. And edge is what
connects these vertices or endpoints.

Fermat's Little Theorem

Fermat's Little Theorem states, if \(p\) is a prime number and \(a\) is an integer, \(a^p\equiv a\) (mod \(p\)). The theorem itself is used and is very helpful when testing numbers to see if they are not prime. It can be easy to see whether a small number is not prime however, with big numbers it can be difficult. One of the most important things to note is that the theorem does not tell whether a number is prime but it does show if a number is not prime. Therefore, the famous theorem also states that if \(p\) does not divide \(a\), then \(a^{p-1}\equiv\) \(1\) (mod \(p\)).

Modular arithmetic is used in discrete math to find remainders. The definition states, if \(a\) and \(b\) are both integers and \(m>0\) then \(a\) is congruent to \(b\) (mod \(m\)) if \(m\) divides \(a-b\). The notion of modular arithmetic deals with the remainders that are found in Euclidean division. The actions of trying to find the remainder is also known as modulo operation or (mod \(n\)) where \(n\) is a an integer. For instance, the division of \(8\) by \(3\) can also be written as \(8\) (mod \(3\)) and we can find the remainder to equal \(2\) thus, \(8\) (mod \(3\)) \(=2\).

Bezout
was a famous mathematician who discovered many beautiful formulas. One of his
most famous theorem/identities dealt with numbers and their greatest common
factors. His theorem states, if integers \(a\) and \(b\) are relatively prime, then
there exists \(x\) and \(y\), integers to satisfy the equation \(ax+by=1\). For any
integers \(a\) and \(b\) excluding zero, let \(d\) be the greatest common divisor such that
\(d\) equals the gcd of \(a\) and \(b\) in other words, \(d\)\(=\)gcd\(\left(a,b\right)\). If this is true then
there must exist integers known as \(x\) and \(y\) to satisfy the equation \(ax+by=d\).

Mathematical induction is used in mathematics to prove many
statements, in particular it is used to prove statements, theorems, or even formulas
that are asserted by all natural numbers. When we say “all” natural numbers it
means any natural number that we may possibly come across on. In order to prove
by mathematical induction, we must first go over some very important rules.

Pascal’s triangle is filled with patterns that can solve
many mathematical problems. One of the neat things that the triangle can do is help with
binomial expressions. First let’s take a look at binomial expressions. Binomial expressions relate to the sum or difference of two
terms such as:

\(x+2\)

\(3x+5\)

\(c-d\)

Stirling numbers of the second kind in discrete mathematics
are used to show numerous combinatoric properties like partitioning a set, (a
number of ways to write an integer of a set), and forming a recurrence relation.

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