This started when my friend Marc LeBrun, who lives in California,
he's a computer scientist. He likes to consider unusual ways of doing arithmetic,
and he came up with this idea.
You know that in school, children have a lot of trouble when they learn arithmetic.
There's all this trouble with the carrys;
it's very complicated.
So he said, let- let's make it
simple for the kiddies. We're not gonna have any carrys.
We're gonna abolish all carrys.
We're gonna change the rules, a bit.
Now when you add two digits, two numbers like 7 and 8, the result is not 15 anymore.
On the moon, it's 8. You pick the bigger of the two numbers.
So 1 plus 1 is 1, sure. 1 + 2 is 2.
When you multiply, you pick the smaller.
So 1 times 2 is 1.
Alright, and that's it. Those are the rules,
that's lunar arithmetic.
(Brady: So 3 times 8 is 3?)
3 times 8 is 3. 3 plus 8 is 8,
you take the bigger of the two.
(Brady: I hope they didn't actually use this on the moon.)
Maybe they didn't, maybe they will
in future.
(Brady: Okay, I understand the rules, I'm a bit worried)
(about what's gonna happen here, but-)
Let's do some arithmetic.
Let's do an addition, for example. Suppose we wanted to add, say, 58 and
19. What you do is you say, 8 plus 9 is 9.
Remember, you take the bigger of the two.
And there are no carrys, 8 plus 9 is 9, 1 plus 5 is
5. You take the bigger of the two, bigger of 1 and 5 is 5.
The biggest of 9 and 8 is 9. So 58 plus 19 is
equal to 59.
That's
lunar addition. What about multiplication? Suppose we want to multiply,
say, 17 by 24.
Remember, digit by digit. So four sevens are?
(Brady: 4) - 4, very good. You pick the smaller. Four 1s are
1. And now two 7s. Two 7s are 2, two 1s are 1. And now we add them up. 1,
1 plus 2 is
2 - you pick the bigger of the two. And you bring down the 4. So 17 times
24 is equal to 124 on the moon.
So now you can do a lot of arithmetic in lunar arithmetic, and you can produce a lot of interesting number sequences.
[Apollo mission recording snippets]
What are the squares in lunar arithmetic?
What's 1 times 1? It's 1.
2 times 2
is equal to - pick the smaller - it's 2.
3 times 3,
3 squared, is 3. 6 squared is 6, 7 squared is 7, 8 squared is 8, 9 squared is 9 - what about 10
squared? 10 times 10 - we have to do it. Zero times zero, you pick the smaller. 0 times 1 is 0.
1 times 0 is 0, 1 times 1 is 1, you add them up. 1, 0,
plus 0 - pick the bigger - and we get 100, just like back on earth! But what about 11 squared?
Well
we have to do it. 1 1 times 1 1, we get 1 1, 1 1 - add them up. 1, 1 plus 1 is 1, 1.
So 11 squared is 1 1 1 - 111.
12 squared: 2 times 2 is is 2, 2 times 1 is 1, 1 times 2 is
1. 1 times 1 is 1 - always pick the smaller. And we add them up 1, 1 plus 1 is
1, 2 is 2; so 12 squared is
112.
And
so on. 19 squared is
119, and so on. So that's the sequence of lunar squares.
[Apollo mission recordings]
If you are interested in the formal details,
this is an associative, commutative,
distributive law - so it works.
We don't have subtraction,
we don't have division, but apart from that we can do anything. What I would like to tell you about next are the lunar primes.
What are the prime numbers on the moon? You're really gonna like this - and I need more paper.
(Brady: I'll get you some)
What are the lunar primes? Well
what is a prime? First of all we have to decide what it means for a number to be a prime.
Now you remember back on Earth,
7 was a prime because you couldn't write 7 as 2 times 3,
or anything except 1 times 7. If the only way to factorise it is as 1 times itself
then it's a prime number. On the moon
is 7 a prime, on the moon? Well, notice
7 is equal to 8 times 7,
right?
Because 8 times 7 - you pick the smaller of the two,
which is 7. So 7 is 8 times 7, so 7 is not a prime!
Because it has a factorisation.
But really before we get down
to the details of this, we have to decide what precisely it means to be a prime. So
formerly a number's a prime if the only factorisation is 1 times itself. What does 1 mean on the moon?
We have to say what is 1 - if I put it in quotes - on the moon?
Well what is 1? 1 has the property that if you multiply anything by 1 you get what you started with.
1 times 3 is 3, 1 times 111 is 111, right. On the moon that's not true.
1 times 3 on the moon is not 3, it's 1. 1 times;
well 1 times 111 happens to be 111 by accident.
1 on the moon - 1 times anything is
basically all 1s and zeros. 1 times 30
is
ten,
because you go through a digit by digit and pick the smaller. So 1 on the moon is not 1.
What is 1 on the moon? Would you care to make a guess?
What's the number? - (Brady: Is it-)
(I haven't thought about it, is it 9?) - It's 9! Very good. Fantastic, brilliant.
Because when you multiply something by 9, it just copies what it sees. 9 times 7 is 7,
9 times 1 is 1 - it just copies what it sees. So on the moon
9 is the 1.
So on the moon a prime number
has the property that the only way you can factorise it is 9 times itself.
So 7 is
obviously not a prime, because seven is equal to 8 times 7.
What about 8? 8 is not a prime because 8 is equal to 8 times 8, so it's not a prime.
It has a factorisation in which one of the fact- neither factor is 9. Any number if- for it to be a prime
the biggest digit has to be a 9. So 25 is not a prime because we could multiply it by 5
and get what we started with.
So we've gotta have a 9 in the number. So 19 is the smallest.
We must have a 9.
(Brady: There's no way to build)
(19 except, what, 19-) - Times 9. - (Right) - You can't factorise because it's got a 9 in it.
So if, if it was 18, we can multiply 18 by an 8 and get 18. We'd have a factorisation.
So you can always multiply by the biggest digit and get what you started with.
So the big- biggest digit had better be 9. (Brady: These primes are going to be rare then?) - Yes, but not all that rare.
So yeah
how rare are they? Well, let's look at the prime numbers. So what about 9 itself? So back on Earth
we don't say 1 is a prime, because it's - we wouldn't have unique factorisation. 2 is the smallest prime on Earth.
But on the moon, just as 1 isn't a prime on on Earth, 9 isn't a prime. But apart from that
we get 19. So the primes, the Lunar primes,
19, 29,
39, 49,
59, 69, 79, 89,
90 is a prime,
91. As long as there's a 9 and there's no obvious way to factorise it;
you have to check that 90 is a prime. We can't get 90 by multiplying two numbers.
You have to check it, but it's true. It's easy to check. 99, we get 109, yes, that's a prime. But
119 is not because, as we saw a moment ago,
19 times 19
is
119. So 119 is not a prime. But roughly speaking most members that have a 9 in them are primes.
So those are the lunar primes. If we were to go to the OEIS and we, in the search window we type lunar
primes - we used to call them dismal primes because teaching of arithmetic in schools were so dismal. It's a dismal definition,
I think we like lunar primes much better. So, well, I can show you the graph - or two graphs. One of the first 200 terms,
technically it's called a pin plot because we draw little pins.
You don't see the pins there because they're quite dense.
Then you get a scatterplot, just a plain graph of however many terms we have, and it looks like we're getting about;
what is it, 2500 terms. There's one more thing I would like to show you.
I'd like to prove that they're infinitely many primes.
You remember- - (Brady: Lunar primes?)
Infinitely many lunar primes!
First of all I'll prove that 109 is prime. And the proof of that is: suppose it wasn't,
it would have to be the product of two two-digit numbers.
They would have to be two digit numbers because of the rules.
So it would have to be ab times cd. But we've got to
get a one at the beginning.
So one of a and c,
remember how it works, the table - the multiplication,
the tableau. If you get 1 at the beginning either a or c has to be a
1. And to get a 9 at the end, both b and d have to be 9s.
That's the other, only possibility because when we do the multiplication
9 times 9 is 9. 9 times 1, well, that's, that's 1 - pick the smaller of the two. Okay,
now we do the c. c times 9 is,
well, it's c. Whatever C is.
Multiplying it by 9 doesn't change it. And c times 1,
well c can't be 0 so it's gonna be 1. Now we add them up and we get 1, 1 plus c. I
don't know what it is. It's the bigger of 1 and c. Well, it's gonna be c.
It could be 1 - anyway what we get is the number 1c9. But c can't be 0,
because if c was 0 we wouldn't get the 1 in front.
So this can't happen. This is a contradiction. If c was equal to 0 we wouldn't get the 1 in front.
We cannot factorise 109 as a product of two primes. And then, likewise,
1 0 0 0 0 9 is prime by the same argument.
There are infinitely many primes.
(Brady: Yeah, no matter how many, you can put as many zeros there as you want?) - As many zeros as you want and you get a prime.
(Brady: Nice proof!) - Yeah. Yeah. Nice theorem. Lunar primes, lunar arithmetic.
[Apollo Mission] Ok Neil we can see you coming down the ladder now
[Preview] ...are the ones that the mathematicians like. These are the primes, or numbers that are related to primes, and
they are - they are elevated
above the ordinary numbers. - 0, 1,
5, 27, 194, 1865 and so on