Hi, yesterday I left you with a question. Do
prime numbers go on forever? Or do they eventually
run out? Did you think of any ideas? Did you
find a way? It is not an easy problem. Because
you cannot just keep checking numbers forever.
So how can you possibly prove something about
an infinite collection? Over 2 ,000 years ago,
a mathematician, Euclid, faced this same question.
Rather than trying to chase primes out to infinity,
he turned the problem around. He said, let's
suppose the opposite. Let's imagine that prime
numbers do run out, that there is a complete
list of all the primes. Maybe it's short, maybe
it's long, but suppose we've found them all.
Now here's the key idea. Take every prime on
that list and multiply them together. It starts
with 2 times 3 times 5 times 7 times 11, and
then it goes to the final number that's in that
set. This answer will equal one gigantic, enormous
number. So every prime number will be a factor
of this ginormous number. Then take that ginormous
number and add 1. Just 1. Now pause and think
about this new number, one bigger than the one
we calculated. We know the original number is
a factor of every prime number. That means this
new number is not divisible by any of the primes
on the list. Because when you divide it by any
of them, there is always a remainder of one.
So it does not fit into any of the rectangles
that's built from the primes that we thought
were complete. Which means either this number
is itself prime or it is built from primes that
are not on our list. Either way, our list was
not complete. And that is the breakthrough. That
is the surprise. No matter how many primes you
think you have, you can always find another.
So, primes do not run out. They can't. There
are infinitely many of them. And what's remarkable
is how this answer comes. Not from checking forever,
but from a simple idea. Assume you're done, and
then show you can't be. From rocks to rectangles
to infinity. A question a child can ask. And
an answer that stretches beyond the horizon.
And if you follow that even just a little, you
have stepped into something amazing. You are
sharing a line of thought with one of the great
mathematical minds in history. Not just knowing
a fact, but understanding why it is true. To
me, that is something to value. Because it means
you can do this. You can learn ideas that at
first feel out of reach. And this is just the
beginning. The beginning of seeing that mathematics
is not just about answers, but about understanding.
And that understanding is available to you. And
that is the beauty of math. It enlightens our
mind beyond what we thought was true.