Hi, yesterday we saw that prime numbers are the
oddballs. They do not fit into rectangles. And
yet, that very feature is what allows them to
become the building blocks of all numbers. And
once you notice something like that, it is almost
impossible not to ask a question. How many primes
are there? How many of these strange, stubborn
numbers actually exist? And what I find fascinating,
is this is not just a mathematician's question.
It is not tied to a certain time or place or
culture. I think it's a human question. In fact,
we've been noticing patterns in numbers for a
very long time. There's an ancient artifact called
the Ishango bone, over 20 ,000 years old, with
markings that many believe reflect an awareness
of prime numbers. Long before modern mathematics,
long before formal proofs, humans were already
noticing that some numbers behaved differently.
The moment you start arranging rocks and realize
that some numbers do not fit into rectangles,
you can ask it. A child can ask it. How many
are there? That is the low floor. But then comes
the next step, trying to answer it. And suddenly
you feel the ceiling because we're not counting
objects on a table. We're talking about numbers
and numbers do not stop. You can always go further.
There's always a bigger number. So now the question
becomes even more interesting. Inside this infinite
world of numbers, does this special group, the
primes, also go on forever? Or... Do they eventually
run out? To me, it is not obvious. You might
imagine that primes become more and more rare
until eventually they disappear. But how would
you ever know? You cannot check forever. You
cannot reach the end of numbers. So how could
you possibly answer a question like this? And
this is what makes it such a beautiful question.
Simple to ask, incredibly difficult to answer.
A true low floor, high ceiling idea. So today,
take some time to consider how would you approach
it? If you had to decide, do primes go on forever
or do they eventually stop? What would you do?
What would you try? Tomorrow, we will look at
a remarkable idea over 2 ,000 years old that
indeed answers this question without ever needing
to reach the end.