Hello!
This is a brief bonus lecture
and what I'm gonna do in this lecture
is show you that there're infinite number
of prime numbers.
In other words,
I'm gonna prove the infinitude of primes.
Now, this is really one of the gems
of the history of mathematics
and the history of science,
and using, as we'll see in a moment,
just primitive tools for the most part
we can prove that, in fact,
there're infinite number
of prime numbers.
And, as I've indicated,
this proof goes back
to someone called Euclid,
who's generally considered
the founder (one of the founders)
of geometry
and the proof dates
to at least 300 BC.
So let's begin with the definition
of what a prime number is.
A prime number
beginning with then number 2
is an integer
greater than or equal to 2
whose only whole number divisors
are 1 and itself.
So, for instance, 2 is the smallest prime.
3 is the next prime
in the sequence of primes.
5 is the next.
7, 11, 13, and so forth.