There are infinitely many primes

Theorem 1.1 (Euclid)   There are infinitely many primes.

Note that this is not obvious. There are completely reasonable rings where it is false, such as

$$R = \left\{\frac{a}{b} : a, b\in\mathbb{Z}\text{ and }\gcd(b,30)=1\right\}$$

There are exactly three primes in $R$, and that's it.

Proof. [Proof of theorem] Suppose not. Let $p_1=2, p_2=3, \ldots, p_n$ be all of the primes. Let

$$N=2\times 3\times 5 \times \cdots \times p_n + 1$$

Then $N\neq 1$ so, as proved in Lecture 2,

$$N = q_1\times q_2 \times \cdots \times q_m$$

with each $q_i$ prime and $m\geq 1$. If $q_1\in\{2,3,5,\ldots,p_n\}$, then $N = q_1 a + 1$, so $q_1\nmid N$, a contradiction. Thus our assumption that $\{2,3,5,\ldots,p_n\}$ are all of the primes is false, which proves that there must be infinitely many primes. $\qedsymbol$

If we were to try a similar proof in $R$, we run into trouble. We would let $N=2\cdot 3\cdot 5 + 1 = 31$, which is a unit, hence not a nontrivial product of primes.

Joke (Lenstra). ``There are infinitely many composite numbers. Proof: Multiply together the first $n$ primes and don't add $1$.''

According to

              http://www.utm.edu/research/primes/largest.html

the largest known prime is

$$p = 2^{6972593}-1,$$

which is a number having over two million¹ decimal digits. Euclid's theorem implies that there definitely is a bigger prime number. However, nobody has yet found it and proved that they are right. In fact, determining whether or not a number is prime is an extremely interesting problem. We will discuss this problem more later.