LAURENCE EAVES: Right.
Well, I was born on May, which is the fifth month, and on the
13th day of May.
Mathematically, that's vaguely interesting because these are
both Pythagorean primes.
Right, OK.
Well, if we look at 5, then we can write 5 as
the sum of two squares.
So 1 squared plus 2 squared is equal to 5.
Now if we take 13--
see if we can work that one out.
Well, 13 is 9 plus 4, so 4 is just 2 squared.
And 3 squared is 9.
So that gives me--
now, the next one, 17.
We can do that as well, because 17 is 4 squared plus 1
squared, which is another Pythagorean prime.
Pythagorean--
we've all heard of Pythagoras, who was a very famous
mathematician.
So why do these numbers bear his name?
Well, if we remember our right-angle triangle.
So we've got a right triangle there, and there's the
hypotenuse.
And Pythagoras tells us that if this thing is of length A,
and this bit here is of length B, then A squared plus B
squared equals C squared.
That's Pythagoras's Theorem.
Now, of course, A and B don't have to be integers.
And C doesn't have to be an integer.
They could be any number.
This could be 3.2 inches.
This could be 5.75 inches--
whatever C is then.
But we can play this game with integers.
And when we do, we generate these Pythagorean primes.
If we look at 13, then it turns out all of these numbers
can be written as 4n plus 1.
So 13 is equal to 4n plus 1, where n is equal to 3.
If we take 17, then 17 is 4n plus 1 where n is equal to 4.
So the Pythagorean primes have this.
We generate them this way, but they've also got this
interesting property.
BRADY HARAN: Can I put any n in there to get a Pythagorean
prime, or only certain n's will do it?
LAURENCE EAVES: No, no.
For example, if we just take the next one in this series,
if we put n equals 5, then n equals 5 gives
me, of course, 21.
OK, so that's just 20 plus 1.
And although 21 is an odd number, it's not a prime
number because it's equal to 7 times 3.
So 21 is not a prime.
It's not even a prime, so it's obviously not
a Pythagorean prime.
So it's not all values of n that give me these, just
special values.
BRADY HARAN: We mentioned at the start that your birthday
is the 13th of the 5th, and they're
both Pythagorean primes.
LAURENCE EAVES: Yes.
BRADY HARAN: Can you remember at what point in your life you
realized your birthday was two Pythagorean primes?
LAURENCE EAVES: When you asked me to talk about numbers.
I have a really favorite Pythagorean prime, which I'd
like to tell you about.
And then I just realized that 13 and 5 are
two nice, small ones.
BRADY HARAN: OK, tell me about your favorite one.
LAURENCE EAVES: Well, my favorite one,
of course, is 137.
BRADY HARAN: For anyone that doesn't know, by the way,
Professor Eaves is obsessed with the number 137.
And you will see him talking about it in multiple videos.
LAURENCE EAVES: Not really that obsessed.
But anyway, the inverse fine-structure constant, which
we use a lot in atomic physics and any problems to do with
the electrons and their quantum behavior.
The fine-structure constant alpha-- its
inverse is almost 137.
Not exactly 137.
And lo and behold, 137 is a nice Pythagorean prime, which
I think is quite nice.
And if we work that one out, of course, we see-- how can we
make it up?
Well, with a little bit of thinking, we can see that 11
squared gives me 121.
Then if I take 4 squared, which is 16,
that adds up to 137.
So can we draw the Pythagorean triangle for that?
It will look something like this.
I've changed the scale of my units here a little bit.
So we've got 11 there, 4 there, and then this diagonal
is the square root of 137.
A nice Pythagorean prime.
BRADY HARAN: So just to be absolutely clear, the
hypotenuse of a right-angle triangle, when the other two
sides are integers, won't be a Pythagorean prime?
LAURENCE EAVES: Its square root is a Pythagorean prime.
BRADY HARAN: Its square root.
LAURENCE EAVES: Yeah.
MATT PARKER:That's 28.
How many arrows do you want?
So the next one, let's say we did 3 to the power of, to the
power of, oh, arrow, arrow, arrow.
Whatever you want to call this.
JAMES GRIME: And that will give you the third perfect
number for--