The Prime Constant - Numberphile

Every prime number is a step on the infinite journey of mathematics, revealing patterns and connections that go beyond simple counting.

I'm going to draw a number line. We're going to need a lot of paper for this, which is why we're running this one on the continuous stream of brown paper. I'm going to do a number line starting from zero. We'll put one maybe here, and I'm going to continue this number line. I will label all the Primes; we'll do them in green as we go along, if that's okay.

So, every single Prime I'm going to mark on this. We may feel like that might not be enough paper; maybe I've got the scale okay. Fine, this quite possibly may be the longest number five video ever. And there they all are. There, that is the 0.4 1 4. Hang on, let me get the rest of these digits: 6 8 2 5 0 9, and maybe more digits: 8511, and it keeps going. There you go, all the Primes.

It's like, you know, Pi is the circle constant, e is the exponential constant, and this is the prime constant. It is a number that represents all the Primes—every single Prime. I'm willing to hear more. Oh, oh, more? Okay, we'll do more. As you, the viewers, can try and predict where we're going with this, I have declared accurately that this is all the Primes.

Now, I'm going to do something seemingly unrelated, and then we will work out how we combine the two. The unrelated thing I'm going to do is writing fractions in base two. Okay, we'll do a half. Now, in base 10, we write a half as 0.5, but then in base two, we write that as 0.1. Then you think, “Well, hang on, how did we extend the idea of base 10?” Because normally when we do binary, it's almost always whole numbers, at least in popular mathematics.

But in reality, if you're doing things with numbers, they're very rarely neat and tidy like an integer; you’ve got to have your decimal places. We can think about how we represent them in base 10, how we do it in base two, and the differences between them. A half is nice and neat in both cases.

Now, some numbers are not necessarily nice and neat. For example, if we wanted to do a third in base 10, that can't be done. It's really annoying, so you end up with 0.333, and then you never stop doing threes—it's threes forever. That three could be done as repeating or something like that, whatever symbol you like to use for that.

In base two, I'm actually going to look it up real quick. Oh, it's also a mess! Here it is: 0 0 1 0 0 1 1 0 0 1 1, and then that repeats. It's like 0 0 1 1 over and over again, right? And that just carries on going, going, going. We often don't think it through in base 10 because it seems so obvious.

But what you're doing is you're taking many decimal places you've got and then dividing it by 10 to the power of how many there are. In this case, and I've not picked particularly great examples, here what you're actually looking at is going to be 5 ID 10 that gives you a half. In this case over here, that's 1 ID 2, which is a half—quite nice.

If we draw a line here, that would be 3 over 10 or 33 over 100 or 333 over 1,000 or 3333 over 10,000, and so on and so on. If we actually ended up with 333 over a 1,000, you can see that's close to a third but not quite. The more we have, the closer we get, but we'll never quite get there. This is the same idea here.

So what we could do is 0 1 0 0 1 1. We're going to stop here, and we divide that by—oh, you know what? I'm going to switch this to base 10 so that in the binary of that...

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Some numbers are neat in one base but messy in another; just like life, not everything translates smoothly.

In this discussion, we explore the concept of fractions and their representation in different bases. We start with a case where we have 1/2, which is quite nice. If we draw a line here, we can express it as 3/10, 33/100, 333/1,000, or 3,333/10,000, and so on. If we actually ended up with 333/1,000, we can see that this is close to a third but not quite. The more we have, the closer we get, but we will never quite reach that point.

This idea can be illustrated further. For example, we can represent a fraction in binary as 0.101011. If we switch this to base 10, we find that it equals 19, and when we divide it by 2 to the power of 6, or 64, we see that it is reasonably close to a third. However, it is not as close as the previous example because this is a larger base, which converges faster. The more we work with these fractions, the closer we can get to our target values.

In a recent video, I discussed a bug in Minecraft related to a strange behavior when boats were falling from very specific heights. The issue arose because the code needed to represent 0.4. In base 10, 0.4 is nice and neat, but in binary, it becomes 0.0110 011, which then repeats indefinitely. This messiness occurs because 5 is a factor of 10, allowing for a neat representation, whereas 5 is not a factor of 2 or any power of 2, leading to complications.

We often complain about base 10 being inadequate for writing fractions due to its factors being only 2 and 5, which yield nice decimal expansions. Base 2 is even worse because 0.4 never terminates in base two. When we convert it into binary code, we must truncate it somewhere, resulting in only an approximation of 0.4. This approximation was the cause of the bug in Minecraft, which I found quite interesting. If you're curious, I will link to that video below for further exploration.

Now, you might wonder where our prime constant comes into play. Let’s convert the prime constant into base two to see its representation. The prime constant can be expressed as 0.0110101001001... If we analyze this, we see that the first position is a zero, positions two and three are ones, while four, six, eight, nine, and ten are zeros. The ones correspond to prime numbers, while the zeros represent non-prime numbers.

Mathematicians have constructed a binary number where there is a one wherever there is a prime and a zero wherever there is not. This construction is completely determined by the fact that there are infinitely many primes, and although they are in a specific order, we have identified them to a certain extent. This creates a very specific real number in base 2.

It is important to note that the base is arbitrary; I used base 10 for ease of understanding. However, it is crucial to assume that this number was created with the knowledge of prime numbers. It is not a method for predicting primes; rather, it is a representation that confirms the existence of primes without providing a means to predict them accurately.

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Every increasing sequence of numbers can be encoded into a single real number between 0 and 1, revealing the hidden beauty of infinite possibilities in mathematics.

In mathematics, there exists a method to represent prime numbers through a specific real number. This is done by assigning a one wherever there is a prime and a zero wherever there is not. This representation is completely determined, illustrating that there are infinitely many primes which are arranged in a neat order. We have discovered these primes to a certain extent and can fill them in accordingly. Thus, this process defines a very specific real number, and while the base used for this number can be arbitrary, it is often presented in base 10 for ease of understanding.

It is important to note that this number was manufactured with the knowledge of the prime numbers; it was not found independently. Consequently, it is completely useless at predicting prime numbers because we only know its value by first identifying the prime numbers and then encoding them. Some might consider this just a novelty, a mere encoding of the primes, but the key point is that one must find the primes first to perform this encoding.

Interestingly, this concept can be applied to any sequence of integers that is always increasing, which in mathematics is referred to as a monotonic sequence. This means that each subsequent number in the sequence is larger than the previous one. If these sequences have already been sorted in numerical order, one can refer to the Online Encyclopedia of Integer Sequences, which contains numerous sequences. While not all of them are monotonic, those that are, such as the primes, can be encoded in a similar fashion.

For example, there exists a Fibonacci constant between 0 and 1 that will encode every single Fibonacci number. However, we must first find the Fibonacci numbers before we can reverse-engineer this constant. Despite this requirement, I believe it does not diminish the beauty of the concept. There are infinitely many and uncountably infinitely many real numbers between 0 and 1, which signifies that there is a vast amount of complexity within this tiny range.

The power of having infinitely many digits after the decimal point is often overlooked. The sequence never ends, allowing for an immense number of values to be packed into this space. This idea extends to any conceivable monotonic series of numbers, which can also be transformed into a real number between 0 and 1. Among these sequences, the prime constant stands out as the most famous, as primes are often considered the mascots of mathematics.

It is particularly intriguing to think about encoding the infinitely many prime numbers, which are notoriously difficult to find, into a single number. This notion is fascinating because it suggests that this number could serve as a signal from an alien civilization. If we were to beam out a sequence like 011, there is no plausible natural event that would generate the primes in this manner. Some might argue that a Game of Life simulation could produce primes, but such occurrences would not happen randomly in the universe.

If one were to encounter this sequence beamed out from space, it would evoke a sense of wonder, akin to discovering the digits of pi. However, one must consider the possibility of different curvatures of spacetime elsewhere in the universe, which could yield a different version of pi—an unsettling thought. In contrast, primes remain consistent; they are always primes. Thus, if we want to communicate our cleverness to the universe, this number would be the ideal choice.

Thank you very much for watching this video. If you enjoyed it and wish to see more of Matt Parker, let's be honest, who wouldn't? You can check out his latest book, Love Triangle, as well as his YouTube channel, Standup Maths, and a playlist of all his previous appearances on Numberphile. I will link to all of that down below in the description and comments, as per usual. Love Triangle is definitely worth a look!