Prime Numbers

So, you want to solve the Riemann Hypothesis?
It can't be that hard right?

There are very few people who deserve the moniker 'genius'.
Without a doubt though, here are three of them:

• Leonhard Euler
• Carl Friedrich Gauss
• Bernhard Riemann

Once you become interested in Prime Numbers you get the feeling that, were we only to know the secret to their distribution we could unlock the universe, predict the future, build time machines and live in houses made of £20 notes.
It's a soul destroying subject though. Essentially we know (pending a million dollar proof) that the primes are distributed on the number line in manner that defies insight.
So.. I should just end this page here really.

But here is a statement that explains the profound nature of this topic:

"Primes are the atoms of multiplication, but we want to know where they sit on the number line — which is an additive structure — and the deepest unsolved problems in mathematics, from the Riemann Hypothesis to the Langlands programme, are all fundamentally asking the same question: how do these two independent operations constrain each other?

Anyway, what follows is essentially just output from my learning about primes and exactly what the Riemann Hypothesis is.
With some pretty pictures, and sounds.

Euler

Let's start with Euler, because why not?
Euler is generally considered the greatest mathematician of all time (the GOAT).
What's more he was apparently a thoroughly decent man.

Euler is perhaps most famous for solving the 'Basel Problem'.
Not only was that work one of the greatest moments in the history of Mathematics, but it formed the cornerstone of Gauss and Riemann's work I'll discuss later.

Euler is also famous for the 'identity': $e^{i\pi }+1=0$ which is famous for it's beauty, but it's importance is that it links together the five fundamental constants.
Also Euler's Formula: $e^{ix}=\cos \left(x\right)+i\cdot \sin \left(x\right)$ which unified exponential functions, trigonometry and complex numbers.

It will become important later in this discussion, but Euler also introduced the notation: √-1 = i
Aside: He wasn't responsible for the word 'imaginary', that was Descartes, but Euler was too polite to correct Descartes.
The point is that Euler essentially laid the foundation for everything discussed below, including along the way, disproving Fermat ($2^{2^n}+1$) and prompting the Goldbach Conjecture.

Very specifically in terms of primes I want to really point to this (Euler's 'lucky number polynomial'):

It doesn't look like much, but this high-school level quadratic predicts 40 consecutive prime numbers.
That might not be blowing your mind right now, but I hope your mind is changed by the end of this page.

Here is plot of Euler's polynomial:
Green dots are primes that the polynomial correctly finds, and red dots are composites (not prime numbers) that the polynomial outputs.
This polynomial never actually diverges from the 'curve' of prime numbers. That is it keeps predicting primes for any value of n.
However the 'hits' become more sparse as n increases.
Try increasing n max using the text entry box under the plot.

To set the tone for the rest of this document here is a prime spiral (see below for more on these spirals) with Euler's polynomial $n^2+n+41$ overlaid. Green dots are polynomial outputs that are prime (hits), red dots are composites (misses), and the dark background dots are the actual primes.
This nicely shows that the polynomial follows the primes, but tracks the primes sparsely with larger values of n.
I'll talk more about prime spirals later.

Before carrying on, let's just take a quick look at this quadratic using the quadratic formula:
Applying the quadratic formula to : $n^2+n+41=0$

where a=1, b=1, c=41:

What's important to notice here is the appearance of the rather magical prime number '163'
This is the largest of the 'Heegner Numbers' (we'll discuss more below).
And this is fundamentally why Eulers quadratic predicts the maximum number of consecutive primes of any polynomial (known so far).
Though Euler didn't (it's believed) fully understand why this polynomial was so unreasonably effective.
In fact nobody fundamentally understood it until 1913 (Rabinowitz's Theorem).

Sums of infinite series

As mentioned, Euler solved the Basel Problem. The Basel problem was simply, does

sum to infinity, or some particular number?
That is, does the sum converge or diverge?

What Euler proved was something unbelievably profound:

That is, Euler proved that the sum of an infinite number of terms is somehow magically related to π
I should pause here for impact, for those who've never seen this before.
But...

What is the sum of the reciprocals of the squared primes?:

And the answer to that is, it's not a neat value like $\frac{{\pi }^2}{6}$, it is approximately 0.4122...

But Euler's proof of the Basel Problem gave the hope that something profound could be found in all infinite sums, even the primes.

These series can be generalised like this
$\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\cdots$
Riemann later named these 'Zeta Functions' (ζ(s))

One thing to note is that you only get an 'answer' (The function only converges) when S > 1.
It kind of stands to reason, because the higher 's' gets, the less each term contributes to the sum.
There's some discussion below about how this can be 'worked around'.

Gauss

Euler was born into a comfortable middle class family, and had access to the Bernoulli family for education.

This does not in any way detract from Euler's genius, but it does explain his rather pleasant demeanour

Gauss on the other hand was born dirt poor. His dad was essentially an odd-job man and apparently quite 'rough'.
His mother was illiterate, but completely devoted to him (Gauss).
Gauss was lucky and was given a Scholarship, it's hard to say where the world would be had he not (as so many don't)

This does not in any way detract from Gauss's genius, but it does explain his un-pleasant demeanour.

By all accounts Gauss was 'a bit of a dick'.

We know (Gauss) that the distribution of prime numbers roughly follows a pattern:

This is known as the 'logarithmic integral function'.
Vastly simplified then, the density of primes near a number t is approximately: $\frac{1}{\ln \left(t\right)}$.
If you draw a curve of the form : $y=\frac{1}{\ln \left(x\right)}$ you get a curve which roughly follows the distribution of primes along the number line.

So you can have a 'rough guess' at what the 'nth' prime is using $p_n\approx n\cdot \ln \left(n\right)$
For example: the 25th prime is approximately $25\cdot \ln \left(25\right)=80.5$, and the actual 25th prime is 97 (Rough!)

This method of estimating primes is not as productive as Euler's quadratic.
But it is the start of a process that eventually uncovers the deeper mysteries of prime distribution.
Where Euler was interested in 'producing' primes, Gauss was more interested in the underlying structure.

Quadratic reciprocity

Gauss was very interested in the factorisation of integers, and in modular arithmetic.
One of things he uncovered is a curious relationship between pairs of primes which would seem to have no other link.
We should start by explaining what Modular Arithmetic is. Think of a clock, it's 13 hours since mid day, what time is it?
It's 1am, you just did modular arithmetic.
A perfect square is a number which equals another number times itself (ie 25 = 5 x 5)

Armed with this knowledge let's look at the next step in the journey (Gauss's 'Golden Theorem' of Quadratic Reciprocity):

It looks scary, lets take an example:
First pick two primes we'll call them p and q, let's go with p=13 and q=17:
Plug them into the right hand side of the equation:

This is an even number so:

Confirming the relationship works both ways:

Or in a more pedestrian manner:
Let's take p, subtract 1 and halve it : 13 - 1 = 12, 12 / 2 = 6
Now lets do the same for q: 17 - 1 = 16, 16 / 2 = 8
In order for these two primes to be 'reciprocal', the product of these two numbers must be 'even': 6 * 8 = 48✓
Are any numbers between 1 and q-1 a perfect square such that, mod q of that perfect square = p?
Working through from 1, we get to 8:
8² = 64, and 64 mod 17 = 13 ✓
Therefore (Gauss tells us) that there must be a perfect square between 1 and p-1 such that, mod p of that perfect square = q mod p.
2² = 4, and 17 mod 13 = 4 ✓

Gaussian integers ℤ[i] and other number systems

So at the age of 18 Gauss had discovered (quadratic reciprocity) that some primes can be expressed as the sum of two squares, and proved it in 5 different ways.
And at the age of 22 he proved 'The fundamental theorem of algebra' which relates to complex numbers and the factorisation of polynomial equations.

Gauss knew that complex numbers with integer values can be used to factorise tricky polynomials. In fact most A'Level students know this:

See how the 'i' terms cancel when we expand the brackets to give us back our quadratic.
The 2ix - 2ix is obvious, and the 4i² because i² = 1

So if a prime can be expressed as the sum of two squares, it can be expressed as a complex product:

We just factorised a prime number!
In the search for a pattern in the primes this is great news surely?
And Gaussian Integers (ℤ[√-1]) are not the only complex number systems:

Let's take the prime number 13:
In ℤ[i]: 13 = (2 + 3i)(2 - 3i) AND
In ℤ[√-3]: 13 = (1 + 2√-3)(1 - 2√-3)
We just split a prime number in two different ways.
This will lead us on to Riemann later.

But first. Gauss also noticed something about composite (not prime) numbers in these complex number systems:
There exists 'The Fundamental Theorem of Arithmetic' (FTA) which states that every integer greater than 1 can be uniquely represented as a product of prime numbers.
Gauss found that these complex number systems can actually break the FTA:

6 = 2 × 3

Or more explicitely for this example:

6 = (2 + 0(√-5)) × (3 + 0(√-5))

But also in ℤ[√-5] we can do this:

Because:

We've factorised a composite using two different sets of irreducible numbers, in the same number system.
In other words, we've broken the FTA using the ℤ[√-5] number system.

But what Gauss found was that FTA is NOT broken in some complex number systems.
He noticed this because of his holistic grasp of complex numbers and modular arithmetic. Because he was a genius.
It is beyond my grasp and requires some heavy duty maths, but Gauss knew that the FTA still holds for ℤ[√-d] with the following values of d:

1, 2, 3, 7, 11, 19, 43, 67 and 163

These numbers are now known as 'The Heegner Numbers' because Heegner proved that these are the ONLY d values for which FTA holds, Gauss only suggested it.
You will have forgotten by now, but remember that 163 popped up on Euler's Lucky Number Polynomial?
163 is the largest, and the last Heegner number.
It is the biggest 'd' for which FTA still holds, and clearly this has some mysterious connection to the primes.

Riemann

By all accounts Riemann was so painfully nice that he was almost disfunctional.
And he was Gauss's protege.

This does not in any way detract from Riemann's genius, but it perhaps does explain his ill-health.

Like Gauss, and Euler before him, Riemann was without doubt a genius.
Most people with even a passing interest in the prime numbers will be aware of the 'Riemann Hypothesis' and the 'Riemann Zeta Function'.
In fact that's probably why you're reading this!
But to non-mathematicians it's very hard to get 'in roads' into exactly what these things are.

Well by now we've covered enough to get at least a fleeting glimpse of what the Riemann Zeta Function and the Riemann Hypothesis are.

Let's just give an analogy first:
Algebra is very powerful, we can substitute a thing we don't know for a letter or symbol, and then work backwards until we know what that symbol means.
Better yet, we can assign a letter to something unknowable, or even impossible and things might work out.
Take:

We cannot calculate the square root of -1, because a minus times a minus is always positive.
But we've seen above, that it works no matter how preposterous it seems.
I mean, I still think negative numbers are preposterous, let alone 'i'

What if we could do something similar in the search for a pattern in the primes.
We could assign a number to the 'mysterious' part, and perhaps it'll just come out in the wash?
This is pretty much what Riemann did, by pulling together the work of Euler and Gauss discussed above in a way only a genius could.

Riemann took Euler's work on the sums of infinite series, specifically:

Which he named ζ(s) [zeta of s].
For example ζ(2) = 1/1² + 1/2² + 1/3² + ... = π²/6 (the Basel Problem from above)

But then Riemann extended the concept such that 's' was a complex number of the form a + bi, where a and b could be any real number (1, 1/2, pi, sqrt(42) etc.)
For example:

In fact this particular example is the first Riemann 'zero'
The 14.1 is shortened for example IRL it is a transcendental number (it's actually more like: 0.5 + 14.134725141734693790...i)
When a complex number results in a 'zero' it is called 'Rho' value. So 0.5 + 14.1..i, is a known Rho value.

So we can see here that calculating the Riemann zeros is extremely difficult.
To get the first one we'd calculate for '0.5 + 10i' and keep increasing (11i etc.) until we get a sign change.
We'd know we'd passed a zero then (we know it's between 14 and 15).
So we'd next do 0.5 + 14.1, 14.2 etc. looking for a sign change.. and so on.
If that kind of trial and error wasn't tedious enough, actually evaluating each iteration of the zeta function is (as you can see) painful.

So what does this tell us about the primes?
Well, it's not good news pattern hunters!

That all the 'real' parts of the Rho values are 0.5 means that the distribution of the primes deviates from the expected log distribution curve by the minimum possible amount.
In other words there's nothing to see there, no pattern, nothing to work with.

As for the imaginary part of the rho values, that's probably even worse.
It's thought that they are all transcendental numbers with absolutely no relationship to each other.
This essentially means that the primes don't interfere with each other over the large scale (they follow the log curve).

The common analogy is that of a sound wave.
The real part of the rho values all being the same means the wave always has the same amplitude, at every frequency.
It's 'normalised' or 'compressed' in music production terminology.

The unrelated nature of the imaginary parts means that the waves never interfere with each other.
That is, the ripples decay before they overlap with other ripples.
Which again means, there's nothing to see.
There's no secret to the distribution of the primes
Just like I said right at the top.

You can see how the 'waves' interact in the interactive chart below.

What then?

The study of the primes is almost as bad for one's mental health as studying the Collatz Conjecture
Our brains are pattern spotting machines, we see human faces in the folds of cloth, and the outlines of continents in the clouds.
The lack of any discernable pattern or underlying rationale disturbs us. Surely, that the primes are distributed in such an inscruitable manner is suspicious in-and-of-itself isn't it?
There just MUST be a pattern?

One obvious direction is to find another method of linking the rho values. That is, finding a link between the transcendentals produced by the Riemann Zeta Zero's.
In fact, just like we know that the primes themselves are roughly distributed along the 1/ln(x) line, so the zeros are distributed along:

You spotted the pi right? Well don't get too excited, it's mainly just an artifact of the Zeta Function.
You have to remember that 'i' means 'rotate in the complex plane'.
However, you're not alone. It's an open question in physics and mathematics as to whether this 2pi has any real world significance.
Perhaps this is an in-road?

Your question should be, are the Rho values distributed with the same impenetrability as the primes themselves, or is there a deeper pattern.
You guessed it, the rho values are distributed 'impenetrably' along γₙ ≈ 2πn / ln(n), in the same way the primes are along 1/ln(x)
Ok.. what else then?

If you threw a stone into the ocean of numbers you would hit a transcendental (Cantor, infinite sets 1874)
Not 2, or 3/16ths or Pi or e or phi.. Some random un-named impenetrable never ending string of digits arcing off into infinity.
The 'algebraic' numbers are countably infinite, the transcendentals are uncountably infinite, so they outnumber the other numbers by an infinite factor.
That is, the chances of a number being non-transcendental are literally zero.
To use the common parlance the primes are a thin dusting of icing sugar on a transcendental cake.
You can cut the number line anywhere and get a single particle of icing sugar, and an infinite amount of cake!

It's unsurprising then that the imaginary part of the Rho values are all transcendentals
Terrance Tao says that trying to prove the Riemann Hypothesis is like trying to climb a mile high sheer cliff face made of polished marble.
There's just no way to start the climb.. No handholds.
Transcendentals are impenetrable. Humanity essentially lacks the tools for dealing with them. Currently (maybe that's your in-road?).
The real start would be in finding a way to prove that ANY given number is transcendental. That would give us an understanding of exactly what transcendence is.

Finally, what are the big boys doing?
Look up Langlands, Hilbert-Pólya, Random matrix theory.
But mostly they're waiting for another Euler.

Visualisation of the Riemann Zeta Function

Below is an interactive reconstruction of π(x) — the prime counting staircase.
The red staircase is the actual count of primes up to x. That is each time we see a prime, we take a step.
So the y axis is 'how many primes so far', and the x axis is just the number line.

The blue dashed line is Li(x) — Gauss's smooth but general estimate.
The green line (when you use the slider) is the result of adding correction terms to Li(x) from the Rho values. (like calculating more coefficients into a huge polynomial)
Use the slider to add correction terms from the zeta zeros (the ρ values). Watch the green curve morph from the smooth Li(x) into the exact staircase as you add more zeros.
Each zero adds a wobble — a frequency in the 'music of the primes'.
Or to be somewhat less poetic, each zero 'encodes' some more information about how the primes are distributed.

Going back to the 'music' analogy, I know if you're anything like me you'll want to know what the waveform that seems to exist on the green line sounds like

To generate this, we subtract Li(x) from the corrected curve to isolate just the oscillatory error term — the wobble caused by the zeta zeros — and treat it as an audio waveform. This clip uses 1,000 zeros sampled over x = 2 to 10,000. Each zero contributes a frequency; the descending pitch reflects the primes thinning out as x grows.

Ramanujan (Honourable Mention)

I didn't add Ramanujan to the list of genius's at the top of the page, not because he wasn't a genius, but because his contribution to the study of primes is not easy to pin down.
Rather like Gauss, Ramanujan was born dirt poor. Unlike Gauss though, Ramanujan's character was more like Riemann's. Shy, polite and considerate.
Sadly, also like Riemann, his health was similarly poor.
In 1914 he noted that e^(π√163) is remarkably close to the integer 262537412640768743.99999999999925 (an in-road?)
You'll notice 163 there again.

You may have (I did so you probably have) wondered how the real part of the Rho values can be 0.5 in the first place?
Right at the top of this page we discussed Euler's zeta functions and we noted that 's' must be greater than 1 for the zeta function to converge.
Well, this is because the methods used to 'solve' the zeta function involves some trickery named 'Analytic Continuation '.
Euler had managed to converge otherwise divergent series, but Riemann was a master at it.

Perhaps the most famous thing that Ramanujan did was the rather famous:

Ramanujan wasn't the first to do this, that was probably Riemann (or even more likely, Euler), but Ramanujan did it in a rather unauthodox way.
Where Riemann had used 'Analytic Continuation' to produce the result, Ramanujan used a unique method (now called Ramanujan Summation).
It was this that caught Hardy's eye in the famous letter.
But what this shows is that Ramanujan was no stranger to sums of series or zeta functions.
Had Ramanujan lived longer it's possible that he might have made progress, such as he was.

Before moving on, and for completeness, here's where that -1/12 actually comes from (it's a factor of 'summation'):

The blue curve shows ζ(s) for real values of s. For s > 1, this is Euler's series $\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\cdots$ For s < 1, it's the analytic continuation. Note the pole (vertical asymptote) at s = 1, the trivial zeros at s = −2, −4, −6, and the value ζ(−1) = −1/12.

And here is where the −1/12 comes from geometrically. The orange staircase is f(n) = n stepping up at each integer. The blue line is the smooth version f(x) = x. The shaded triangles are the difference — each has area ½. Naively, infinitely many of these sum to infinity. But the Euler-Maclaurin formula regularises this infinite sum of differences, and the regularised total is −1/12.

Heegner Numbers (a small pointless excursion)

We've already seen that Euler created a polynomial which predicts prime numbers : $n^2+n+41$
In fact it turns out (after I had done what I'm describing below), that Euler's polynomial predicts the maximum possible number of consecutive primes, and then just keeps on giving.
But I wondered if somehow encoding ALL the Heegner numbers into a polynomial might somehow produce even better results.
To this end I (well, Kiro) curve fit an 8th degree polynomial to exactly fit the 9 Heegner numbers.
The intention was to hopefully find something 'beautiful' in the coefficients of the polynomial that might inspire further investigation.

$h\left(n\right)=\frac{107}{20160}{n}^8-\frac{247}{1260}{n}^7+\frac{4349}{1440}{n}^6-\frac{9077}{360}{n}^5+\frac{357349}{2880}{n}^4-\frac{131843}{360}{n}^3+\frac{3169631}{5040}{n}^2-\frac{238711}{420}n+205$

Not so beautiful, but we're not done yet.
Because this polynomial was constructed using Lagrange interpolation we can do something interesting which is to pull out factors which are 'factorial' numbers:

$h\left(n\right)=\frac{107}{8!/2}{n}^8-\frac{247}{7!/4}{n}^7+\frac{4349}{6!\times 2}{n}^6-\frac{9077}{6!/2}{n}^5+\frac{357349}{6!\times 4}{n}^4-\frac{131843}{6!/2}{n}^3+\frac{3169631}{7!}{n}^2-\frac{238711}{7!/12}n+205$

Not sure this made it any more beautiful, let's try jiggering things around to pull out only 8!

$h\left(n\right)=\frac{1}{8!}(214n^8-7904n^7+121772n^6-1016624n^5+5002886n^4-14766416n^3+25357048n^2-22916256n+8265600)$

This seems like it might reduce further?
The highest common factor of the coefficients is 2, which gives us:

$h\left(n\right)=\frac{2}{8!}(107n^8-3952n^7+60886n^6-508312n^5+2501443n^4-7383208n^3+12678524n^2-11458128n+4132800)$

Again, there's nothing really jumping out here even in its most reduced form.

Anyway, here's a calculator to evaluate h(n) for any value of n.
If you enter an integer, you'll get back an integer, and spefically, if you enter 1 to 9 you'll get back the Heegner numbers.
But you can enter floating point numbers, or integers beyond 9 etc. Some of which MAY return integers
For example try :

• 8.6676704617
• 8.6848179921
• 8.7167728556
• 8.7317220894
• 8.7598518480
• 8.8439367674

I plotted graphs, charts.. performed many many types of analysis on the polynomial output and in the end I realised that the polynomial contained no magic at all.. but here it is for your amusement.
It doesn't beat Euler's $n^2+n+41$, unsurprisingly.

Prime Number Spirals

Humans are visual creatures, perhaps if we can just SEE the problem with our eyes, we can spot a pattern?
Whilst bored one day I watched a youtube video about prime number spirals.
Being a programmer of sorts I decided to knock up a bit of code to draw these myself. I started with just replicating prime spirals.

You do this by plotting the prime numbers in ever wider circles. So we'd take the first prime (1) and we'd draw it somewhere in a circle which was 1 unit away from the centre.
Aside: yes, I know 1 isn't prime — but for the purposes of plotting spirals it doesn't matter to me. Where on that circle? Well a circle is 2 PI radians around (about 6.3), so if we were plotting the number 1, it'd be about 1/6th around the circle (sort of 2 o'clockish).

The next prime (2) we draw in a circle '2' units from the centre (because it's the second prime number, so we draw it about 1/3rd around the circle.
And so on.. the prime number 5, we draw in the 4th shell (it's the 4th prime number) and it's at about 10 o'clock on the circle (2 pi / 5) etc.

Because all primes are odd numbers (except 2) they'll all end in a 1,3,7 or 9, so we can colour the dots accordingly.
This is what it look like at varying levels of zoom.

My big idea was this.. What's the point in using 2 PI radians. Why not make a circle '10' around? What would happen if we did that? I mean, we count in base 10. In an ideal world 'll see straight lines, giving me a chance to be able to create some sort of equation to predict prime numbers, bask in glory and piles of money.
This is what happens when you make a circle '10' units around instead of '2P' units around :

OMG! 'm RICH!!!

So I sat there, staring at the screen for a few minutes thinking, it's very unlikely that I have discovered something that Gauss, Fermat or Euler have missed, so what's going on?

What happens if I make a circle some other arbitrary number around, let's try 7.

No, this is a mess in comparison. The colours are all mixed, and the ideal number of 'rays' is 4. One for each possible last digit (1,3,7,9).
What's the lowest number I can make the circle and still get 4 clean rays?

Ok, so it turns out a circle of 5 units around gives me 4 clean rays. Large numbers give me too many rays, or rays with mixed colours. What is it about 5 that's special? Well, 5 is the most interesting number. If I can be bothered 'll do a post about Roger Penrose and 5 fold symetry and how it links to the Golden Ratio etc.

But it turns out 5 is not interesting per se.
What's going on here is that all the primes effectively get 'mod 5ed'.
That's why we get 4 clean rays. But hey, we're a step closer aren't we?

Now we know that all prime numbers can be drawn on lines with polar angle (2pi / 5) * ending (about 1.2566371 radians for numbers ending in a 1 etc.).

So we can calculate the angle in polar coordinates. If we can also calculate the radius, we can use simple Pythagorean geometry to predict prime numbers, and we'll be immortal!.
How do we calculate the radius? Well.. this is where it gets complicated 🙂

Notice that the lines in the mod pictures above have random gaps in them. I need to find out if there is any pattern to those gaps.
NOTE : Of course.. this is the whole point isn't it!

Here is another sonification — this time encoding the last digit of each prime as a musical note. (see the rays above)
The first 500 consecutive primes are mapped to a CMaj7 chord: primes ending in 1 play C, ending in 3 play E, ending in 7 play G, and ending in 9 play B. Each note sustains for 20ms before the next prime takes over. The melody you hear is dictated entirely by the sequence of prime endings.

It sounds like a sci-fi film, largely because so many have actually used this method of generating sound effects!

What if, instead of plotting the prime numbers themselves, I plot the gap between primes. So we take the first 5 primes 1,2,3,5,7, the gaps are 1,1,2,2 (hey, there's a pattern! He says with his tongue in his cheek.)

So what I did here was divide the prime number by the previous prime, then multiply by a large number to get numbers in the range of 1 to 2 (or 10 to 20 etc.) ie 29/23=1.26086956522 * 10 = 12.6.. etc.
We know we can do this because of Erdős's (another extremely interesting individual) proof of Bertrand's postulate (lolz :))
This gives us a way of getting a 'gap' which doesn't grow as the size of the primes grow. I'm not going to go into much detail about the results except to say you should click on the X1000 photo and know that we live in the matrix.

Here is a sonification of those normalised gaps. Each consecutive prime ratio p(n)/p(n-1) is mapped to a pitch — ratio near 1 (small gap) gives a low tone, larger ratios give higher tones. You can hear the early primes as a brief rush of high frequencies, then it settles into a low drone as the ratios compress toward 1. The occasional upward pings are the larger gaps.

It seems that at large scale, the gap between primes does seem to give us some sort of pattern. But I'm not sure if these 'artifacts' are related to how I'm doing the calculations or if they really exist.

At x10000 we can see 39 rays. That is, the gaps between primes appear to fall into roughly 40 discrete sizes. Have I discovered something, or is this more stating the obvious?

These plots are enigmatic. We think we can see patterns here, and we can.. But what do these patterns actually mean?
It's a lot like the Mandelbrot set. We can see patterns in the Mandelbrot set, but we don't know what they mean, and we don't know if they have any meaning at all.
Perhaps if Euler had access to a computer, he would have known what these patterns mean. But me.. no.

In the next section I'll try and superimpose patterns onto these spirals and be rich!

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