This web page is an expanded version of a presentation given in Boston at the 7th International Conference on Complex Systems

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Prime Numbers are the Holes Behind Complex Composite Patterns Visualizing Number Theory with the Divisor Plot Jeffrey Ventrella Jeffrey@Ventrella.com (www.Ventrella.com)

1 Introduction

"Prime numbers are what is left when you have taken all the patterns away."
- Mark Haddon, The Curious Incident of the Dog in the Night-Time , page 12

When I was young, I got a large sheet of paper and a pencil, and started plotting a graph, like the one shown at the right. Each number on the top-row has its divisors plotted underneath, and they are plotted vertically according to their sizes. After plotting many numbers, I began to see patterns.

According to Gregory Chaitin, “A concept is only as good as the theorems that it leads to!…instead of primes, perhaps we should be concerned with the opposite, with the maximally divisible numbers!'' [Chaitin, 2005]. I read this statement in Chaitin's book 'MetaMath', several decades after making that pencil drawing. I remembered the patterns, and the unanswered questions. Having only gotten as far as the introduction of the book, I dropped it and ran to my computer. I decided to return to this drawing and explore it in more depth, so I created an interactive computer visualization (link provided above) which allows exploration of much larger numbers, with a greater range of divisors. Consider the original drawing: zoom out to see a bigger field of numbers, and then replace the numbers with dots - more patterns can be seen.

For example, the divisors < 120 for numbers near 151,080 are shown above. Why do these patterns seem complicated - almost random? Especially considering the simplicity of the rule that is used to plot them. They exhibit apparent randomness, yes - but also some obvious order. Perfect fodder for the pattern-hungry human brain. That is what this exploration is about - a quest to understand what causes these patterns. Integer, Pattern, and Human Intelligence Gazing at these dots scattered haphazardly across a black background (as I prefer to render them), I feel like an ancient astronomer trying to find some order in the chaos. I don't expect to see Chariots, Big Bears, or Scorpions. But interesting patterns are indeed popping out all over the place, the more I look. Karl Sabbagh believes that if there is intelligent life elsewhere in the universe, it will be capable of numerical counting. "The stars in the sky are discrete points and cry out to be counted by intelligent beings throughout the universe (at least the ones who can see)." [Sabbagh, 2006]. Counting naturally leads to truths such as "2+2=4", as well as all integer math, primes. etc. This is an example of an argument against mathematics being a purely human invention. And unlike some such arguments, it is not overly Platonic, metaphysical, or theological. Leopold Kronecker sez: "God created the integers; the rest is man's doing." We have magnificent occipital lobes, and our brains are wired to find patterns, at all levels - consiously and unconsciously. My exploration of the divisor plot begins with my visual pattern-seeking brain - and progressively applies language and math to my discoveries. I believe this is the natural trajectory of human mathematics.

Overlapping Patterns Thomasson [2001], Wolfram [2002], and others, have generated small variations of this graph to illustrate the distribution of primes. We hear of primes described as the “building blocks” of all numbers. Let's turn that concept on its head. Instead, let's think of primes as the negative spaces behind complex objects. Imagine a series of picket fences stacked in front of each other. Each picket fence has different spacing between its wooden slats.

The superimposition of fences creates line moire patterns. Consider the Sieve of Eratosthenes , a process which involves repeatedly hopping along the number line, with increasingly larger steps, progressively stamping out the composite numbers, to identify the primes. It is like stacking these picket fences, each one with a larger gap between its slats - to eliminate the holes. Some holes will always remain. Those are the primes.

A Definition I call this graph the “divisor plot”. It is equivalent to the set of all integer locations in the x,y plane (where y is positive) for which x mod y = 0. Let us call these integer locations "divisors". As integer locations, these divisors lie on a 2D lattice with cells of size 1. The y coordinate of each location is a divisor of x. For every integer x on the number line, there are two or more numbers that x can be evenly divided into (its divisors). Prime numbers have only 1 and themselves as divisors. Composite numbers have these plus others. The number of divisors of an integer x (the divisor function ) is denoted as d(x). Let us refer to the actual set of the divisors of x as Dx. As an example, D6 = {1, 2, 3, 6}. Every Dx is unique – related to the fundamental theorem of arithmetic.

Relation to the Divisor Function If you plot a graph of the divisor function across the number line, you will notice that it is quite jaggy. This jagginess is related to the distribution of primes. This graph can be thought of as a vertical stacking of the divisors in the divisor plot. Sometimes, the way to understand something that is seemingly random is to see it as ground instead of figure, or as the shadow of something in a higher dimension.

Showing Prime Factorization Another way to visualize numbers is to plot only their prime factors, and to use brighter colors if they have higher multiplicity, as shown here. For example, the prime factors of 12 are 2 x 2 x 3 (or 22 x 3). Therefore, 2 and 3 are plotted under the 12, and the 2 is twice as bright as the 3. Since 16 = 24, it is plotted with a 2 which is four times as bright. One problem with this technique is that most of the variation in brightness gets pushed up towards the top of the plot, where the small primes are, making it hard to see any patterns resulting from differences in brightness, as shown below. However, some intriguing wavy patterns do show up among the larger (darker) primes. This is something to explore further.

Since I am interested in exploring all of the structure of integer divisibility, I have decided not to use the technique shown above for this exploration, and instead to plot every divisor D. Stereo In fact, there is so much to see when the divisor plot is shown full-throttle - it can even pop out in stereo revealing some interesting subtle patterns. Below is a view that an astute pattern-finder named Jeff Rossman found while using the Java Applet. He was exploring the fifth perfect number (33550336), and his eyes accidentally crossed, and the patterns crossed over and locked in: his brain landed upon a new dimension! If you are skilled at viewing stereo images in this way, give it a try: cross your eyes so that the green and red symbols come together. Then, with your eyes locked in at that focal point, slowly raise your view to the dots in the upper half. It takes a while, and for some people it never comes together. But if it does, you will see that the field of dots becomes a deep sloping field. The slight "bumps" (differences in dot "height") might be due to the fact that Jeff compressed the x axis (increasing the number range within view). Aside from this artifact, you can think of your stereoscopic vision as a way to probe the periodicity across multiples - along the "rows" R. This is explained in the next section. Let us now proceed and look at some of the many patterns in the divisor plot. Let's move onto section 2.