Barking Up the Prime Number Tree
You’ve heard it before:
“Prime numbers are the building blocks of all numbers”.
“Prime numbers are like atoms”.
I think prime numbers are more like HOLES – peeking out from behind the patterns created by the composite numbers. Yea, prime numbers may be atoms when it comes to multiplication. But atoms in themselves are not so interesting. It might be more productive to study the molecules. The composite numbers generate a wealth of patterns – including the prime number sequence. Consider:
We often hear the question, “Why have we never found a formula that generates only prime numbers?”
Answer: There probably is no formula. Stop looking at the prime numbers. Mathematician Gregory Chaitin suggests that there is a lot more to be learned from studying the composite numbers.
It’s like this. What if you were in a dense forest with the sun shining high above. You set out on a quest to figure out a way to re-create the exact pattern of light speckles on the forest floor. These speckles of light are caused by the sun passing through leaves and branches. You will have a hard time finding any algorithm, any 3D computer graphic program…any way…to represent that pattern of light. That is because you are looking in the wrong place. Look at the trees. They are making shadows. Represent the trees (and the way the light passes over them and lands on the ground) and you will have a way to generate the shadows and therefore the light speckles. It’s like using the Sieve of Eratosthenes to cast shadows.
Primes are leftover cracks between the composites. They are the empty, pattern-void, gaps hiding in the crevices of something wonderful indeed: The collection of composite numbers, with all their symmetries, family relationships, and overlapping patterns. They form some of the most beautiful structure I have ever seen.
Primes are cool, of course. Don’t get me wrong. In fact, Cicadas use them. Cicadas have mating cycles in which they all emerge from pubescence en masse, engage in a huge orgy, and then die. The “Periodical Cicadas” do this every 7, 13, or 17 years. Why did they evolve to have prime number mating cycles? Why of course: to HIDE from their predators. These predators would love nothing more than to descend on an orgy of cicadas en masse, for a periodic feast. But in order to do this, the predators would have to evolve foraging cycles that are synchronized to the mating cycles of the cicadas.
My point: These little buzzing critters have hit upon the value of primes as a way to evade structure – as a way to hide behind the usual polyrhythms of the biosphere. Indeed, primes are what is left when the patterns are removed (which is why they are so useful in cryptography).
As cool as primes are, it’s a waste of time trying to identify the heartbeat of the primes. The heartbeat has been fibrillating since the beginning of time, and it shows no sign of stopping.
In fact, the way I see it, the sequence of primes is NOT a heartbeat. It is all the silent gaps between an infinity of heartbeats – all beating at different frequencies. These are the composites.
So check out the composite numbers. I’m talking numbers like 12! (12 factorial). Among the big composites you will find symmetries within symmetries. Maybe even some metaphors or hints to the structure of the universe. There’s lots of material for visual patterning when visualizing the Sieve of Eratosthenes for big numbers like nine billion. And if you’re like me, you’ll gobble it up – and maybe some mathematical insights will pop out in the process.
If you can’t see the forest for the trees, look at the trees.