Our base-ten numeral system represents numbers by taking powers of ten, multiplying them by various factors, and adding the results together: for example, 936 = 9 × 10² + 3 × 10¹ + 6 × 10⁰. This is good for practical calculations, but mathematically, it’s a rather arbitrary way to represent numbers. There’s nothing inherently special about ten; it’s just the number of fingers most people happen to have.

A more natural way to represent an integer is its prime factorization, a set of prime numbers raised to various powers and multiplied together: 936 = 2³ × 3² × 13¹. This is like breaking a number down into its atomic parts: for every number, there is exactly one way to break it down into primes, and this factorization gives some fundamental information about how the number behaves in various mathematical contexts.

When we write a number’s prime factorization as we did above, we still end up using base ten. In that previous example, 2³ × 3² × 13¹, the “13” means 1 × 10¹ + 3 × 10⁰, and even the other single-digit primes and exponents are ultimately written in base 10. Recursive Prime Numeral System is a way of writing a prime factorization without any hint of base ten or any other base: everything is written using primes.

My first thought in designing this system was that all the numbers making up the prime factorization could themselves broken down into factors: for example, if a factorization contains 13⁴, the power 4 can be written as 2². However, the 13 can’t be broken down any further into factors — that’s what it means to be prime.

But if we represent 13 using its index — 6, as it’s the 6th prime number — then we get something that can be broken down further: 6 = 2¹ × 3¹. We can continue this process by representing 2 and 3 using their indices as well:

The 2 in that expression can be broken down in the same way:

This cannot be broken down any further, as the only numbers remaining are ones.

Now that we have this expression for 6, we can use it to write 13:

This becomes ungainly pretty quickly, so it would be nice to have a better way to represent this information: a form of visual notation that clarifies the structure of this kind of expression. This notation will need to represent a few things:

• The number 1
• The multiplication operation: A × B
• Another operation: the A-th prime to the power of B
• Some way of nesting operations, like parentheses

I have come up with several ways of doing this.

Dots & Lines

One possibility is to use horizontal and vertical lines to represent the two types of operations. Let’s write A × B as A B , and “the A-th prime to the power of B” as A B . (Of course this looks like division, but in this context it represents something else.) Using this notation, we can write 13 in a much more concise way, which makes it easier to see the structure of the expression:

The sizes of horizontal bars indicates how the parts are nested:

So you see a horizontal bar, its inputs come from the spaces above and below the bar, up until you hit any larger bars (if you do). For example, the bar highlighted in blue here has its inputs from the shaded region:

It is possible to make this system even more concise by not writing the ones at all. If a horizontal bar has nothing in the space above it, then the first input (prime index) is assumed to be 1; if it has nothing in the space below it, the second input (exponent) is assumed to be 1. So that above example becomes:

For aesthetic purposes, I’ll replace the shortest horizontal lines (which represent 2) with dots: .

To write 1 on its own, you can write a multiplication bar with nothing on either side. As with horizontal bars, the blank spaces where inputs would be are assumed to be ones, so represents 1 × 1, which is 1.

Here’s all the numbers from 1 through 1024:

It can look fun to color the parts differently:

Rectangles

(Explanation coming soon)