First blog post?
I don't really have much to post here for now, I will start adding things here sometime later
hopefully. By the way, did you know that there are
infinitely many prime numbers? You probably did, but I'll prove it below anyway, since I have
nothing better to do. The following proof for the infinitude of primes is due to Euclid (300 B.C.):
Let $S = \{p_1, \dots ,p_n\}$ be an arbitrary finite set of $n \in \mathbb{N}$ primes. Define
$$P := \prod_{i = 1}^{n} p_i$$
and, $q := P + 1$. Observe that $q > p_k ~ \forall p_k \in S \Rightarrow q \notin S$. Now, there are
two possible cases.
Case 1: $q$ is prime, in which case we conclude that there's a prime which is not
in the original set $S$.
Case 2: $q$ is not prime. This implies there's a prime $p$ that divides $q$. Let's
assume $p \in S$. Then, $p$ must also divide $P$, and hence must also divide $q - P = 1$. Since no
prime divides 1, we have arrived at a contradiction. Hence, $p \notin S$. Again, we have shown that
there's a prime number that is not in the original set $S$.
Hence, we have shown that for any arbitrary finite set of primes $S$, $\exists p \in $ such that $p$ is
prime, but $p \notin S$. Therefore, there must be infinitely many prime numbers. This concludes our
proof.
A really simple but cool proof. Also, now I know that MathJax is working correctly.
Bye :)
