The Metaphysics of the Prime Numbers and Why They Do Not Make Music

Many years ago a girlfriend mentioned her suspicion that the prime numbers are triggers of change in the evolution of the cosmos. Neither of us were mathematicians, or even much good at arithmetic, and it took me a minute or two to make even a little sense of this idea. It seemed interesting, however, so I suggested she try it out on an online mathematics forum. I’m not sure she ever forgave me for this suggestion. It took her a few weeks to recover from the experience. This sparked an interest in the primes and their metaphysical significance.

Back in 1969, at the end of the first year of my ‘A’ level Mathematics course, I was transferred to Technical Drawing, which tells you all you need to know about my skills is this department, but seemingly complicated problems may often be reduced to quite simple ones by approaching them from an unusual angle with a naïve and simple-minded attitude, and I felt I could do this.

As a musician of sorts I immediately noticed and read the popular best-seller The Music of the Primes by Marcus du Sautoy. It may seem hard to believe, but by the end of this book I still believed the prime numbers were mysterious and unpredictable. However, having learned that there is a $1,000,000 prize for a proof of the Riemann Hypothesis, which is all about the behaviour of the primes, I set off to investigate number theory.

Three years later I was still unable to understand the Riemann Hypothesis and gave up trying, but I learned a lot along the way and perhaps someone might find it interesting. This is all very naïve, so any mathematicians here should probably look away now.

The Behaviour of the Primes  

At first glance the primes appear to occur quite randomly on the number line.  

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 …

In order to explain this sequence we need to examine the mechanism that produces it.

Imagine what happens if we add the prime numbers one at a time to the number line, starting with an empty line. Each prime will produce a sequence of products that extends to infinity.   

0 and 1 produce no products, but when we add 2, the first prime, we create the infinite sequence of the powers of 2 (4, 8, 16, 32, .,.). Even so, the number line remains approximately empty.

When we add the second prime 3, however, all hell breaks loose. The products of 2 and 3 form a combination wave that ‘beats’ on every multiple of 6, or at 6n where n is any number. (6, 12, 18, 24, 30, 36 …). 

This combination wave of 2 and 3 has a frequency of 6 and its shape ensures that four out of every six numbers are not prime. As a musical pattern it is a bar of 6/8 with a quaver note on 1,3,4,5 and rests on 2 and 6.

If you want to hear this rhythm then here’s a good trick. Tap your hands on your knees to the rhythm of the repeated words ‘nice cup of tea’ using them in the order Together, Right, Left, Right; Together, R.L.R; …. Now you are playing 2 in the bar with your left hand and three in the bar with your right. A useful trick for musicians.

This combination wave repeats to infinity, ensuring that primes greater than 3 occur only at 6n+/-1. If you look again at the sequence of primes this pattern may now be visible.  

When we add the third prime 5, this creates a combination wave with a frequency of 30. The shape of this wave ensures that no more than 8 in every 30 numbers is prime, a little less than 2 in every 6, since now there cannot be a prime at 30n+/-5. (Thus 25 and 35 are not prime).

When we add 7 we create a combination wave with a frequency of 210. And so on to infinity and beyond.

From this we can see that the primes do not create music. Rather, they are the missing frequencies in the harmonic spectrum of the products of the primes, which starts out as something like music but soon become pink noise as we add successive primes. It never becomes white noise, since we know there are infinitely many primes.

It turns out that this sort of harmonic analysis has some use. Using Excel I was able to produce a prime number checker that worked up to the limit of the software. Then, using little more arithmetic than is described above, I managed to produce a proof of the Twin Primes Conjecture. (The conjecture that there are infinitely many twin primes). Feeling rather excited I sent it to a mathematician for a comment. His verdict was that it was a correct heuristic proof, but not rigorous. That is to say, it correctly described the mechanism that produces the distribution of primes and shows that it should produce an infinite quantity of twin primes at 6n+/-1, but it does not show that there can never be a glitch in the system so does not, unfortunately, qualify for a major mathematics prize.  Still, it seemed interesting that such a proof could be made by using only simple arithmetic.

There is an intimate relationship between metaphysics and the foundations of mathematics, but whether the primes have any metaphysical significance I cannot say. Still, if the world is made of vibrations and oscillations, as quantum field theory, string theory and modern models of the early universe seem to suggest, then it does not seem unreasonable to suppose that the evolution of the combination wave we are examining here, which is determined by the primes and is transformed each time we add a new prime to the mix, has some connection with the evolution of the cosmos. The nature of the connection is beyond my imagination, but after some time delving into the issues I came to the conclusion that my girlfriend, (the intense barrage of flak she received when she mentioned it on a mathematics forum notwithstanding), had had a quite reasonable idea.

 Postscript

A few days after writing this post and posting it on my Substack page I stumbled upon a recent video on Youtube discussing the discoveries of the James Webb telescope, in the course of which which it is reported that the prime number sequence has been found in an AI analysis of the cosmic microwave background radiation.

The video commentary suggests that there is something astonishing about this discovery and speculates that intelligent beings might be using the primes to communicate. I don’t know why. My naive analysis suggests that the primes are very likely to appear in the data, and that to interpret them as alien communications is no more justified than interpreting them as music.

Here’s a link to the video. The relevant part of the report starts at 5’15”.

https://youtu.be/fm0lY4niQhY?si=6UBtmNKuV2dq0MDy