So I’ve got 1/4, 1/3, and 1/2. If I just look at those three I can say, okay, what is the sum of all the denominators?
It’s a little bit like Pascal’s triangle or it’s in that kind of similar vein where we’re going to have these rows of numbers and each row depends on what happened in the previous. In the first row I just write 1, 1. Then for the next row every time an adjacent pair of numbers adds up to 2 I’m going to insert that in between them. So I’m going to take a 2 and kind of put it in between those two 1s. And so for the third row every time two numbers add up to 3 and they’re next to each other I’ll insert that 3. So this will look like 1; 3, 2, 3, 1. For the next one I insert a 4 whenever adjacent numbers add to 4; so I’ll have 1, 4, 3, 4, 2, 4, 3, 4, 1. For good measure I’ll just do one more. Between the 1 and the 5 that adds up to 6, none of the next adjacent ones seem to add up to 6 so the only other one I have to add is at the end. So this grows by a smaller amount.
Now someone comes along and they’re just asking questions about this, what are the patterns I can observe in this Pascal-esque pyramid; and if they just count how many elements are there in each row they’d say this one has two numbers in its row, this one has three numbers, this one here has five numbers, seven, eleven, thirteen. And so you’re compelled to ask, okay how- is this- Prime numbers! Yeah and, you know, I could say is there any reason that we should expect prime numbers from this? You know is- this is just a completely random tower, you know is there any reason we might expect prime numbers? And I could say, actually this isn’t as random a tower as you might think, it’s related to these things called Farey sequences, where if I want to write out what are all of the rational numbers where the biggest possible denominator is 1 that sits somewhere in the interval from 0 to 1 there’s really only two rational numbers available - I can have 0 which I could write a 0 over 1 or 1, which is 1 over 1. And if I say, okay what are all the rational numbers in that interval? You know, kind of the unit interval from 0 to 1 with a denominator that is at most 2. And you can say, well I’ve got those those two that I already had, but now could also include 1/2 into those. And so you know kind of might mark 1/2. And what if we allow the denominator to get as big as 3? And he said, well in that case I can have all of the ones that I previously had but also 1/3 is available, and 1/2, and also 2/3 is available. And I’m writing them in the order that they would show up on the number line so I’m kind of sorting them from least to biggest. You can show actually that this sequence here of this Farey sequence, all of the rational numbers maximized by a certain denominator, is exactly the same as what we were doing up here. All of the denominators on the bottom correspond to one of the rows up here. And there’s a kind of cute reason for it, but when you have a sequence of three different rational numbers in this Farey sequence- so I’ll just do one more row to kind of see like where does 1/4 or the fourths fit in. We’ve got 0 over 1, 1/4 is going to fit in here between that and 1/3 then we’ve got 1/2, then 1/3 and then 3/4 comes up there, and then 1. If we just look at three adjacent elements - so maybe a take these here. So I’ve got 1/4, 1/3, and 1/2. If I just look at those three I can say, okay, what is the sum of all the denominators? It’s 4 + 3 + 2 = 9. Adding up the numerators and denominators of two fractions will give you the mediant of the two fractions. This is like what someone who hasn’t learned about fractions might think adding two fractions is. This mediant is always between the two fractions and in Farey sequences it will always be the one that sits in between.
The pyramid pattern is related to rational numbers in the sense that the denominators must add up for a certain number to be introduced. For example, when adding 1/5, 2/5, 3/5, 4/5, none of these fractions reduce, so 5 must be a denominator.
The pattern does break, however, as 17 is left out of the club. To continue the pattern, one must ask how many reduced fractions there are with an 8 and with a 9. The number of numerators that don’t share any divisors with a denominator is called the Euler totient function. We can see an example of Richard Guy’s strong law of small numbers in the first few numbers we look at. For instance, when we add 6 to 2, we get 8, and when we add 4 to that, we get 12, which is divisible by 3. If we add 6 again, we get 18, which is divisible by 3 and 2. When we add 4 to this, we get 22, which again is divisible by 2 and 3. Finally, when we add 6 to this, we get 29, which is a prime number. This pattern is not necessarily representative of larger numbers–for instance, there are only 25% primes between 0 and 100. So, it’s important to understand that small numbers are not necessarily representative of larger numbers, and that patterns found in small numbers may not hold for larger numbers. Check out our podcast with Grant! A link to it can be found in the description, as well as more videos featuring Grant and other content you may enjoy.