Imagine a magic rope. It's long, stretchy, and beautiful. Let's take it with us on a trip through the positive integers. We can start wherever you like, and we'll go toward infinity. We put one tip of the magic rope down at our feet, at whichever integer we happen to be standing on. The other end of the magic rope trails behind us on the number line on a lower integer. Along its length, our magic rope contains numbers that are full of possibilities.
Although we're still wandering around, letting the rope drag freely along the numbers, we might be curious about the prime numbers our magic rope happens to be touching. Primes, remember, are numbers that are only divisible by themselves and 1, and they are one of the greatest mathematical mysteries. Primes are to integers (whole numbers) as the elements are to chemistry. Just as all matter is made up of the elements on the periodic table, so all integers are made up of primes. Any whole number that's not a prime can be created by multiplying primes together - and there's only one way to create any given whole number by doing so!
For example, the first four primes are 2, 3, 5, and 7. If we multiply them together, we can produce 4 (2x2), 6 (2x3), 8 (2x2x2), 9 (3x3), 10 (2x5), 12 (2x2x3), 14 (2x7), 15 (3x5), 16 (2x2x2x2), 18 (2x3x3), and so on. So we've been able to create most of the numbers from 2-18 by multiplying just our first four primes. And we could create many more numbers - an infinite number, in fact - by multiplying just these four primes an increasing number of times. But even between 2 and 18, we have a few holes. We couldn't create 11, 13, or 17 in this way, because these numbers are primes themselves.
That's why primes are a little like the chemical elements. And just as establishing and studying the chemical elements was a crucial step in the development of chemistry, so understanding the patterns in the prime numbers is central to mathematics. So we might be curious about what we can find out about the primes on our magic rope.
Once we decide where the far tip of the rope will go, it would be especially nice if we could say exactly what prime numbers are on our magic rope. Unfortunately, that would be a little bit like figuring out exactly who the love of your life will be, years before meeting him or her. You might be able to predict that you'll fall in love one day, and maybe even that you'll get married. You might know some things about that person: their gender, the way they'd treat you, that sort of thing. But it's hard to be more specific than that.
We have the very same issue for the prime numbers. It's hard to know their exact identity, especially when there are so many of them. After all, they go on forever! Figuring out the exact identities of the prime numbers on our magic rope is going to be a challenge, when there are so many primes, and yet so many more non-prime numbers!
Just like our one true love, though, we might be able to predict something about the prime numbers on our magic rope, depending on how far we stretch it out behind us. Perhaps we can identify the smallest or the largest prime in our magic rope, or at least figure out what their maximum or minimum size could possibly be. Alternatively, maybe we can say exactly how many primes are on it. Or if even that is too exact, we might be able to find a cap on the number of primes that could possibly be on it. There might also be at least a certain minimum number of primes on our magic rope.
Let's take another look at our magic rope. We're standing right now on, say, 192. We put the magic rope with one tip at our feet, right on 192. The other end is on some lower number. Could be 10, or 79, or 191. We don't know how big our magic rope is. Is it always the same size? Does it get bigger or smaller as we go higher?
Notice that we now have two general questions.
- What can we say about the number of primes on our magic rope?
- How big is our magic rope, and where are we standing?
These two questions must be related. Primes are spaced out in a curious fashion. As we go higher and higher, they tend to be spaced out further apart. Remember, the first few primes are 2, 3, 5, and 7 each separated by a distance of 1 or 2 from its neighbors. Higher up, we find 1153, 1163, 1171, and 1181, which are spaced out by distances of 8 or 10. These spacings get wider and wider as we go higher.
However, even some very large primes are pretty friendly: there are an infinite number of primes that are no farther apart from their neighbors than 246, and it's possible that there may be an infinite number of primes that are as close as they can possibly be - separated by a distance of only 2. We call these "twin primes," and this hypothesis is the "twin prime conjecture." So primes in general like to have more and more space to themselves the higher we go, but there also seem to be some primes that are quite cozy with their neighbors.
So the size of our magic rope is going to matter when we're trying to figure out what we can say about the number of primes on it. For example, if our magic rope is of the size 246, we can say that as we go higher and higher, we'll never stop finding places where both tips of the magic rope are touching prime numbers.
But notice in this most recent example that the rule only applies on certain steps as we walk up the number line. And those steps might be rare indeed, the higher we go. We're guaranteed to never stop finding them, as long as we keep climbing, but it will be a rare day when we do. The size of the magic rope isn't the only thing that matters. Any relationship between our magic rope and the primes will depend on whether we're trying to find a narrow rule, that only applies on certain steps, or a universal rule, that applies on every single integer as we walk from our starting point toward infinity. The twin prime conjecture is a narrow rule.
We are finally ready to understand Ramanujan primes. These special primes play a crucial role in a universal rule about prime numbers. As we walk from any starting point upward towards infinity, it turns out that we can say something about the primes on our magic rope for every step along the way, if we think about the size of our magic rope and about our starting point in a certain way.
The magic rope needs to get bigger the higher we go. In fact, we can be more precise, and say exactly how big it must be, the higher we go. For whichever integer we're standing on, our magic rope has one end at our feet, and the other end at an integer that's just shy of half the size. If we're standing on an even number, we put the other end of the magic rope on a number half the size of the number we're stepping on, plus one. 10 goes to 6, 20 goes to 11, 100 goes to 51. If we're standing on an odd number, we put the magic rope's other tip on a number half the size of the number we're standing on, rounded up. 11 goes to 6, 21 goes to 11, 101 goes to 51. See how the size of the magic rope gets bigger as we stand on higher and higher integers?
If this is how we define the magic rope, then we are able to learn one of the holy mysteries of the primes. It was revealed to the Indian mathematician Ramanujan by the goddess Lakshmi, and is a thought of God (according to him).
We can't say exactly which prime numbers live on our magic rope, or how big they are, or precisely how many there are. But we can put a lower bound on the number of primes on our magic rope. We can say that there must be at least a certain minimum number of primes. Some magic ropes have at least one prime. Others have at least two. Our magic rope may have at least 1,000 primes inside of it, or perhaps even more! This lower bound depends on where on the number line we're standing, which also determines how big our magic rope is.
Can we be more specific? There is a lower bound to the number of primes on our magic rope. But can we say exactly what that lower bound is, depending on where we're standing on the number line? As it turns out, we can.
If we know the size of this lower bound on a certain integer, then we can be certain that it will either stay the same or get bigger as we climb higher and higher. This isn't obvious. Remember that as we climb higher, the primes get more spaced out, making it harder to fit them all on our magic rope. At the same time, as we climb higher, our magic rope is getting bigger, making it easier to fit more numbers on it. But at the same time, the lower end of the magic rope, which is in a region of the number rope where there are more primes, is getting pulled along with us into a region of lower "prime density."
Which of these trends will win out? Will the primes get so spaced out that they stop fitting on our magic rope? Or will our magic rope get so big that it can easily fit more and more primes? Lakshmi revealed to Ramanujan that the second possibility is true: the magic rope gets bigger so fast that its growth outpaces the increased spacing of the primes.
That tells us something important about our magic rope and the lower bound of the number of primes that live on it. If our lower bound is 10 when we're standing at a certain point, it means that there are at least 10 prime numbers, and maybe more, on our magic rope. Not only that, it means that no matter how high we climb, we can always be sure that there are at least 10 primes on our magic rope! Of course, the lower bound may not always be 10 - this is just an example. If we do know what our lower bound is at a certain point in the number rope, then we can be quite sure that the lower bound is the same or greater for numbers even further toward infinity on the number line.
But how high do we have to climb before we can be certain that our lower bound is 10?, or 2, or 10,000? As we climb higher, does the lower bound increase by one at a time, or does it jump up by different amounts?
As it turns out, Ramanujan was able to show that as we climb higher, the lower bound stays steady for a while, and then occasionally increases by 1. When we're standing on the number 2, for example, our magic rope has at least one prime on it. One tip is on 2, and the other tip is on 2 ÷ 2 = 1. 2 is prime, but 1 is not, so our magic rope contains just one prime, which is two. We happen to know that when we're standing at 2, the magic rope contains precisely one prime.
However, Lakshmi isn't guaranteeing this. The magic rope only guarantees that there's at least one prime here. When we're standing at 10, for example, the magic rope stretches from 10 down to 5. Two primes live on this magic rope, 5 and 7. That fits our rule that there should be at least 1 prime on the magic rope.
This lower bound stays the same until we get to 11. Here, the lower bound increases by 1. At 11 and beyond, there are always at least two prime numbers on our magic rope! At 17, the lower bound increases by 1 again, to 3, so that there are always at least three prime numbers on our magic rope once we get to 17.
You might have noticed that 2, 11, and 17 are prime numbers. This brings us to our last point. But first, let's review what we know.
- As we climb higher through the integers, our magic rope gets bigger, so that one tip is at our feet. To find the other end, go to a number half the size of the number we're standing on now, and either round up (if we're at an odd number and get a fraction when dividing by 2) or add 1 (if we're at an even number and get a whole number when dividing by 2).
- There is always a lower bound to the number of primes on our magic rope.
- The size of that lower bound starts at 1. As we climb higher through the integers, the lower bound either stays steady or increases by 1.
It would be really nice to know when the lower bound increases. As it turns out, the lower bound increases on some, but not all, prime numbers. 2, 11, and 17 are the first three points where the lower bound increases. These are called Ramanujan primes.
Here is how they are defined mathematically. From Wikipedia:
The nth Ramanujan prime is the least integer Rn for which π(x) - π(x/2) ≥ n, for all x ≥ Rn.
Can we relate this crisp, formal, and perhaps intimidating definition to what we just talked about? Let's break it into pieces:
"The nth Ramanujan prime"
As we walk toward infinity, these are the numbers where the lower bound on the number of primes on our magic rope increases by 1. The first of these Ramanujan primes is the 1st, the second one is the 2nd, and we might say that an arbitrary one is the "nth."
"is the least integer Rn"
Remember, once the lower bound on the number of primes on our magic rope increases, it will stay at least that high to matter how much higher we go toward infinity. We're curious to find the very first, smallest point where the lower bound reaches a certain level - which is just another way of saying that we're looking for the point where the lower bound increases by 1!
"for which π(x) - π(x/2) ≥ n."
π, it turns out, is not just a number. It is also a function, which is kind of like a factory. You put raw materials into the factory, like steel and rubber, and it spits out a useful product, like a car. Likewise, if you put a number into a function, it'll spit out another useful number.
Let's consider another magic rope, which we'll call the enchanted rope.
Again, we put one tip of this rope at our feet. But while we put the other end of the magic rope at a number a little over half the number we're standing on, such as 10 and 6, or 499 and 250, the other end of the enchanted rope always goes on the number 1. The magic of the π or "prime-counting" function is that it tells us the exact number of primes on the enchanted rope!
Earlier, we considered all the things we might want to know about primes on our magic rope. One of them was the lower bound, which we can find on our magic rope. The enchanted rope has more specific information - not just the minimum number of primes there might be, but the exact number of primes there are! It's the difference between knowing that at least 3 of your friends coming to your birthday party, and knowing that exactly 5 of them are coming.
Why bother with the magic rope and Ramanujan primes, then, if we have the π function to tell us exactly how many primes are on the enchanted rope? The most important reason is that, although we have a useful estimate, we don't know how to precisely compute the π function! It's really just a word for a mathematical invention that hasn't been created yet - the cold fusion reactor or the flying car of math.
However, the π function is still useful in order to express certain thoughts, just like the word "flying car" might help spark an inventor's imagination. In this case, π(x) - π(x/2) is just a way of expressing the exact number of primes on our magic rope. Remember, we don't know the exact number. We only know the lower bound.
That's where the next bit comes in. When it says π(x) - π(x/2) ≥ n, it's saying that the exact number of primes on our magic rope is at least equal to n - so n is our lower bound for the number of primes on our magic rope! And the Ramanujan primes are those special primes where n increases by 1.
"for all x ≥ Rn."
Rn, remember, is our Ramanujan prime - the number where our lower bound n increases by 1. This bit is just saying that the lower bound is equal to n not only exactly at the Ramanujan prime number, but also for all the numbers ("for all x") higher than this (Rn).
If you were a mathematician, you might be curious to know more about Ramanujan primes. Remember that prime numbers are a great mystery. We don't know how to calculate exactly how many primes are on our magic rope, and we certainly can't easily compute the locations of the prime numbers as we climb toward infinity. We can only find new ones with a combination of extremely complicated algorithms and a supercomputer. But remember, there are an infinite number of primes. Finding gigantic unknown primes is a very expensive endeavor unless we find a better strategy.
To eat away at the problem bit by bit, you might try to find new things to say about Ramanujan primes - again, the points where the lower bound of the number of primes on our magic rope increases by 1. If you can't say exactly where these Ramanujan primes are, perhaps you can figure out another magic rope - let's call it the Lakshmi rope - where they live. In fact, a few mathematicians by the names of Sondow, Laishram, Nicholson, and Noe revealed the Lakshmi rope in 2009-2011. Perhaps there are further mysteries to be revealed inside of the Lakshmi rope itself.
I composed this article for a few reasons. First, I wanted to understand what a Ramanujan prime was for myself. It helps me to think about mathematical problems in a visual, or geometric, sense. As I tried to apply a little imagination to that terse formal definition of Ramanujan primes, I found that the contemplation was so beautiful that I wanted to try and share it. And as I wrote the article, I found that the act of writing it uncovered what felt like the first sparks of a creative relationship with pure mathematics. Translating a visual impression into crisp algebraic terms, and then contemplating other associated questions about the topic and their relationships, gave me a little bit of an idea about what it might be like to do mathematical research.
This, it seems to me, is exactly the sort of relationship with mathematics that the very best math teachers hope to inspire in their students. Today, I was able to inspire it in myself. And I hope to pass some of that onto you.
Thanks to jmv, gjm, and Taleuntum for catching mathematical errors!
A correction and a philosophical quibble.
The correction: You say
but this is wholly false; for instance, the number 123 is not a prime (it's 3x41) and is not the sum of two primes (it can't be the sum of two odd primes since it's odd itself; the only even prime is 2, and the thing it's the-sum-of-2-and is 121, which is a square and therefore not a prime).
There's a famous conjecture (not proven as yet, though there was some exciting progress a couple of years back) saying that any even number from 4 up is the sum of two primes, but that's not the same thing.
The philosophical quibble: Then you say, after talking a bit more about making integers by adding up primes,
which seems quite wrong to me; there is an analogy between the chemical elements and the prime numbers, but it's about multiplication (every positive integer is the product of primes in essentially one way), not about addition (typically numbers are the sums of primes in many many ways).
Thanks for the catch! I'll put in a correction and credit you at the bottom.
I didn't know about Ramanujan primes and I enjoyed this article about them, so thanks!
One small note: I was a bit confused about whether the magic line's endpoints are included in the magic line. Initially, I thought they both are, but the formal definition says that the lower endpoint is not included. Maybe others were also confused.
That's a great question! Let's think about it.
Let's say we have 10 primes at or below x, and 6 primes at or below x/2. That means that there are at least 4 primes (10 - 6) on our magic line. The lower point can include one of the primes "at or below" it. So one of the 6 primes at or below the lower endpoint of the magic line (as I originally defined it as "half the starting point, rounded up" - it's changed now) could be located right on the endpoint. If that was included as one of the primes on the magic line, then there would have to be 5 primes on the magic line - a contradiction. So no, I think the lower endpoint must not be included. I fixed the post by altering the definition of the lower endpoint of the magic line and credited you at the end.
This example doesn't fit the updated definition:
Good read, I don't think I'd heard of Ramanujan primes before.
Thanks :D I’ll update that soon.
I don't think this is correct. I think that the closest is Goldbach's conjecture, which is for even numbers and is unproven.
I think 27 is the first counter example. 27-2 is 25, which is not a prime. 27-(any prime other than 2) is even, so not a prime.
I think of primes as analogous to elements because of factorisation, not summation.
A bit later in the essay, just before the introduction of twin primes, 9 is included in the list of primes.
Thanks, gjm pointed that out too! I'll credit you as well.
It looks as if jimv got there before me, in fact. (I hadn't seen his comment when I wrote mine. Sometimes some time elapses between when I open a page and when I actually read its contents, and sometimes I forget that I ought to refresh before commenting...)
There's still this slip (the erroneous 9) in the text:
Thank you so much for proofreading!
You say there is a lower bound on the number of primes on the line, but in fact there are many. Any number not greater than the full number of primes is a valid lower bound. What you are looking for is the greatest lower bound that is also a lower bound for all higher/longer magic lines.
I was very confused when you were talking about counting primes before you mentioned the rule for the line length. If you'd started by explaining how long the line is (which is kind of arbitrary), then the discussion of the lower bound for primes would have been easier to follow.
Thanks for the suggestions! What you're illustrating here with your comments actually gets at the heart of what I was trying to accomplish here.
The formal definition of a Ramanujan prime starts (and ends) with the most specific statement definition possible. In a math class or textbook, you might start with the definition, and then unpack the parts. In my opinion, this starts with confusion - a terse riddle that may be intimidating - and only gradually move into more familiar terrain. I wanted to try moving in the opposite direction, by talking about relatively familiar concepts and gradually making them more specific.
Clearly, for you at least, this reversal, or the way I executed it, made things more confusing rather than less. That's an unfortunate outcome, but I still feel excited about continuing to try this approach with other math concepts.
You're absolutely right about your nitpick, and I thought about that while writing, but ultimately decided to leave that bit of nuance out. It's important, but my aim was to put a visual and intuitive way of thinking about Ramanujan primes into the reader's head, from which they can more easily "recompute" the bits of nuance that the article doesn't convey - or extract them from further conversations and reading in a more formal context. Just as we can't expect a mathematical treatise that is optimizing for specificity and accuracy to also achieve a high level of intuitiveness and engagement, I decided to sacrifice some specificity in order to achieve greater intuitiveness. But I expect that for a mathematically sophisticated audience, this may not be ideal.
I would have found the explanation much clearer if you stated at the outset whether the length of the line is chosen by us or for us. As is, the explanation has a lot of parts that are allowed to move from the outset: the location where we are standing, the length of the line extending behind us, and the number of primes on that line. Since the goal is to impose a relationship between these three components, and ultimately our standing location is the only bit that is allowed to vary, it would have helped me make things more concrete if you started with something like "Wherever we plant our feet, the magic line extends backwards towards 0 and then whispers a number in our ear, hinting at the primes it contains."
This sets up the three key questions for the theorem: how far does the line extend backwards, what number does it whisper, and what does that number have to do with primes?
This is a beautiful sentence. I wish I'd thought of it :D I'll see if I can find my own version, because it strikes a lovely balance between leaving the specific details for later in the article, while building anticipation for them.
Feel free to use it wholesale.
Relatedly, I was horribly confused when I started reading the article because I didn't understand that the "magic line" wasn't actually a line in the mathematical sense (lines go on forever, line segments don't). For me it would have been a lot better to call it a magic (piece of) rope.
That’s a great idea, might steal it and update the post! Thanks for the suggestion.
This is the main thing I am going to retain from this post (sorry, Lakshmi).
Until now, if asked to prove why there are infinite prime numbers, I would have answered "because Euclid's Theorem" (Euclid's proof is very easy to remember and explain verbally without formal notation). It never occurred to me that I could have just answered "because there are infinite integers, duh".
What really shocks me is that I've just spent 20 minutes digging "infinite primes" results, and no one points out this. Every "infinite prime numbers" page I found reports either Euclid's Theorem or something even more complicated (it seems that people still publish 1-page papers proving the infinitude of primes). I'm very confused. I mean, "infinite integers imply infinite primes" is not a fully-fledged formal proof, but it shouldn't be difficult to write an actual formal proof from that statement... am I failing to see some obvious things? (like, maybe that proof boils down to Euclid's Theorem?)
I was actually wrong about this!!! After reading your comment, I thought about it more. We can create an infinite number of integers by exponentiating a single prime, like 2, an arbitrary number of times. So we could do 2^2, 2^(2^2), 2^(2^(2^2))), etc. Of course, we can easily find numbers, like 3, that can't be created in this manner. But without a proof, it's not immediately obvious that we couldn't create all the integers by multiplying a finite set of primes an arbitrary number of times. Even if we've proven than every integer greater than 1 has a unique prime factorization, we need a separate proof to show that infinite primes are needed to construct all integers greater than one, rather than just one more prime than we currently have in the set. This is the proof that Euclid's Theorem provides.
The fact that there are infinite integers can only let us deduce that there are infinite primes if we've already proved that every integer greater than 1 has a unique prime factorization (the Fundamental Theorem of Algebra). This point might be a little less obvious :)