Hi, yesterday we saw something remarkable. Prime
numbers never run out. No matter how far you
go, there will always be another. So, a natural
question follows. If primes go on forever, where
are they? How often do they appear? At first,
you might expect a pattern. After all, these
numbers are not random. They are completely determined.
A number is either prime or it isn't. There is
no guesswork. And yet, when you start listing
them, 2, 3, 5, 7, 11, 13, they do not feel like
a pattern. Sometimes they're close together,
11 and 13. Other times they spread out. There's
a gap between 23 and 29. And as numbers get larger,
those gaps tend to grow. Primes become more rare.
So now, we're in a rather interesting place.
We know they go on forever, but we do not see
a simple rule telling us exactly where the next
one will be. It's like looking at something that
is perfectly ordered, but appears almost random.
And that tension has fascinated mathematicians.
for centuries. Over time, we've learned something
that, at least to me, is surprising. Even though
we cannot predict exactly where the next prime
will be, we can begin to understand how often
they appear. Not exactly, but approximately.
As numbers grow larger, primes thin out in a
very consistent way. There's a kind of hidden
rhythm to them. Not a simple pattern you can
see at a glance, but a deeper structure that
emerges over time. So primes live in this strange
space. Completely determined, yet difficult to
predict. Structured, yet mysterious. And maybe
that is part of their beauty. Not everything
that is real is immediately obvious. Sometimes
understanding does not come from finding a simple
rule, but from learning how to see differently,
from stepping back, from gaining altitude. So
today, as you think about prime numbers, hold
on to that tension. Something can be fully determined
and yet still feel unpredictable. And tomorrow,
we will go one step further. and look at just
how much we still do not understand.