Hi everyone. My name is Meni, and I’m here
for the #MegaFavNumbers project. I’ll present
to you one of my favorite numbers over 1 million.
This isn’t the kind of thing I usually do,
but they – the actual math Youtubers – made
it look like just about anyone can join. So
here I am.
Before we get to the number I actually chose,
let’s start with this one.
This is a very interesting number in its own
right. Maybe someone else will make a video
about it. Some of you might recognize it,
or at least have a guess as to what it is.
This number is none other than 2 to the power
of 32. Where 32 is itself a power of 2.
This number plays a key role in computing,
in programming. If you have a 32-bit variable,
this is the number of possible values it can
take. If you have memory addresses that are
32 bits wide, you can’t address more than
this many bytes.
If you were to mine some bitcoins, you have
a nonce which is a 32-bit value. So you can
only go over this many hashes before you have
to change something other than the nonce.
If you take this number, and multiply it by
the current difficulty of the Bitcoin network,
you get the total number of hashes calculated
every 10 minutes.
But this is not what I’m here to talk about.
Let’s see what happens when we take this
number, and add 1 to it.
This number gave quite a bit of trouble to
the mathematician Pierre de Fermat.
He was actually a lawyer in his dayjob, but
he did a lot of math. Today, Fermat is most
famous for the so called “Fermat’s Last
Theorem”. But, regardless of that, he was
also interested in numbers of the form 2 to
the power of 2 to the power of n, plus 1.
The first number of this type is simply 3,
which is a prime number.
The next is 5, also prime.
17 is also prime.
257 is prime.
It’s not so obvious, but 65537 is also a
prime. At this point, Fermat thought, “maybe
all numbers of this form are prime”.
But then we have this number.
I talked about how a number similar to this
one is important in computing. But back then,
they didn’t have computers. So when Fermat
tried to verify whether this number is prime,
he had some trouble doing so. In fact, he
died still thinking that this number – and
all the numbers that follow – are prime.
Only in 1732 Euler came along and proved that
this number is composite. It is not a prime
number. It can be factored like this:
Today, we have computers, and we have better
understanding of number theory. So we checked
many of the next numbers in the sequence,
and all of them were composite. Maybe there
is another prime down the line, but we couldn’t
find it. So when Fermat thought all those
numbers are prime, he wasn't just wrong, he
was very, very wrong.
And the numbers which proved him wrong, are
these two numbers over here. Finding them
was hard; but once we have them, it’s very
easy to multiply them and see that they are,
indeed, factors of this one.
In fact, we don’t even need both numbers.
If we only have one of them, it is very easy
to divide and find the other. So really, either
one of these was enough to prove Fermat wrong.
The number 641 is less than 1 million, so
by the rules of the contest, it is disqualified.
This makes the other number – 6700417 – the
winner by default.
Default? Woo-hoo! The two sweetest words in English language! De fault de fault de OWWW!
There is a lot more information about Fermat
primes on Wikipedia, so make sure to check
that out.
This video was part of the #MegaFavNumbers
project. If you have a favorite number over
1 million, make a video about it, include
the hashtag #MegaFavNumbers, and do it by
September 2.
There are lots of cool videos in this project.
Make sure you watch the video by 3 blue 1
brown – which, much like the Hunchback of
Notre dame, is a tale of a man, and a monster.
Thank you. Enjoy the rest of your day.
