Anyone who has delved into the math and sciences has been introduced to the problem of finding prime numbers. It is one of the holy grails of mathematics and is featured along with the Riemann Hypothesis, Goldbach’s Conjecture, and several others.
With a little spare time and little curiosity many math, enthusiasts explore the problem with the drive to poke around more seriously. Unfortunately, too much free time with prime numbers and you might as well be flirting with madness.
Lately, I’ve been flirting with madness. The problem seems simple enough. How hard could it be to find numbers WE invented and along a straight line of infinite length? That “infinity” might be a hassle though.
The Prime Number Puzzle, Concisely
It's great that mathematicians have made the prime search widely available to the public but finding them with fancy algorithms and or visualizations is still a very surface-level understanding of the problem and the “grail” we are hoping to find. So, what is the challenge, and what glory could possibly have a massive impact on all of mathematics and the disciplines that touch it?
The Challenge: The ultimate puzzle with primes is finding a pattern in how they are listed along the Real Number Line.
At the moment, we don’t know how to predict where to find a prime number along our number line. It may seem trivial but the difficulty of this problem is as grandiose as its implications for a solution. The glory of finding a pattern could result in a profound simplification in the expressiveness and calculation of our mathematical solutions.
Since primes occur much less frequently than other numbers, they act as a sort of doorway to a unique and indexable location. One of the most valuable outcomes of finding primes would be having a unique index to store and retrieve information along infinity which would make computational searches, maps, and jumps much quicker. Think prime factorization!
Brilliant minds have been trying to crack this puzzle for ages so It's easy to find yourself thinking that it’s impossible. It's even easier to think impossible when other people say it is. But, in all my years of unsuccessful attempts at things I have always come out on top with new knowledge. Win or not. Plus, I’ve personally never heard the universe tell me anything was impossible. As far as I know, anything is either possible or your guiding question stops making sense. Speaking of which, let’s double-check our guiding question.
Does Our Prime Question Make Sense?
Question: In what pattern are primes listed along the Real line and can that pattern be uniquely indexed?
Unfortunately no, our guiding question doesn’t make complete sense. The problem isn’t very obvious either. It took traversing 2000 years of mathematical history to finally get a hold of it which had me reading the oldest copy of Euclid’s Elements I could find, as well as multiplication tables on clay tablets written by ancient Babylonians.
As a result, I’ve narrowed down the problem to a conceptual conflict in how our modern numbers are formulated. All of us today use the “Real Numbers” to do any sort of calculations. But, there are some backward compatibility issues between the Real Numbers we use today and earlier number systems including the Natural Numbers we think Euclid used. The biggest issue is this:
Problem: Unlike Natural Numbers, Real Numbers are not “listable” and thus can’t be indexable. Not yet anyways…
Real numbers are not listable because their cardinality, a measure of a Set’s size, is somewhat adjustable. The Cardinality of the Real numbers are assumed to be strictly larger than any more primitive set such as the Natural numbers but verification has been elusive. Interestingly, it’s not a secret at all that Real numbers are not listable, we already know that there is no one-to-one correspondence and or function that can map Real numbers to Natural numbers. Mathematician Georg Cantor spent quite a bit of his life trying to prove what later became the Continuum Hypothesis, CH, or that there IS NO set between Real numbers and the Natural numbers.
Let’s summarize that last point
- We know there is no one-to-one correspondence between Real numbers and a more primitive set.
- We don’t know for sure if there can’t be one-to-one correspondence between Real numbers and a more primitive set
In this way, the jury is still out. At this point, I can only speculate as to why Cantor stressed himself over this so much but I think it will become more obvious as to why as we move forward.
In Set theory, Real Numbers are uncountably infinite and thus are larger than any more primitive set as shown in the figure below.
In the figure above, right to left, we have (N) Natural Numbers, (Z) Integers, (Q) Rational Numbers, and finally (R) Real Numbers. Each number system is larger as you move to the left and includes the set before it as listed above.
It is worth noting that some of these incompatible features of Real Numbers, like being uncountably infinite, gave modern mathematics superpowers and helped bring us all the technological advancement of the modern age. This is especially true with calculus. One benefit of Real Numbers is the ability to establish a limit anywhere, which is very calculus friendly. Thank you Real Numbers…
√2 and the Distribution of Primes
Prime numbers are Natural numbers. Since √2 is strictly a Real number, it can’t be a prime like the other primes we have. However, √2 IS a Real number and real numbers are what we use for everything we do now. I think you will find it interesting how fundamental √2 is for finding primes within our Real number system.
Looking at the figure on the left we see prime numbers (left) listed with composite numbers (right). Prime numbers are divisible by one and itself only. If you imagine this list getting much longer then the primes listed on the left would have blue dots that would extend out and grow in a triangular fashion. Unfortunately, their growth shape doesn’t seem to be connected to their listing interval as of yet.
The Prime-Counting Function, π(x), shown below, has helped many mathematicians understand the basic distribution of primes and serves as a good starting point for amateurs. You’ll notice that taking the prime numbers above and laying them on the x-axis also gives a triangular shape.
In number theory, the prime counting function counts up when it encounters a prime number. The vertical shifts upwards are where the function hits a prime number and the plateaus are composite numbers after a prime. From this, we get the prime number theorem.
Interestingly, along the line y=x, any product x*y =1. It makes sense that the prime number theorem and prime counting function look a lot like the primes I have listed along y=x shown below.
If we start with the base of the green triangle which is 2, you can see that the bases of the triangles following each prime match the prime counting function exactly. However, I’ve confirmed only the first handful through to 31.
The concentric circles you see radiating from the center in the photo above are radii that correspond to the first few prime numbers (except one which is there for reference and is grey). This view is nice because prime numbers on the Y-axis cooccur with primes listed on the X-axis at the same rate on y=x. This is where we find our proposed prime number in full force.
Notice that the first 3 linear lengths along y=x are 1.4142….etc. If you type √2 into a calculator, this is exactly what you should get. Along this line, we can see that √2 seems to be a multiple of every prime number. For example, if you take the green and blue triangles and apply the Pythagorean theorem you get the following.
I’ve checked up further and the trend is consistent in that all these distances are a multiple of √2. I included 1 there to show that √2 seems to also be an alternative definition of one which would make it a factor of every Real number. This would include other curious numbers like pi, Euler's number, and the golden ratio. I am still wrapping my head around this one. Could 1 and √2 both be factors of every Real number? What would that mean? I’ll set that down here.
This isn’t too striking though considering our unit circle is based on a unit area equal to 1. Looking back now it is a bit strange that a cross-sectional area would be given the definition of 1 rather than or in addition to as surface area but we’ll save the physical implication for later. However much you or myself might hate the idea of √2 being a prime number, it could replace our prime number of 2 in the Real Number system. That would give us √2, 3, 5, and 7.
Regardless, just using the x-axis to find primes is quite a bit trickier as shown in the graphic below.
This figure was made by rotating down the red triangle so that its hypotenuse is right along the X-axis. You’ll notice here all these numbers have now become irrational. I used the graph of the natural-log(x-2)+√2 to help line up that center point and get it as close as possible but the algebraic fit is lost when approximating √2.
I think we can see now that finding primes will mean considering √2 as some factor. However, this is just part of our pattern-finding mission. There is still a bit more to this if we want to explore the full problem.
Prime Quest Update: Where Do We Stand?
Let’s revisit what we’ve done so far. Ultimately, we need a pattern for listing and indexing primes along the Real number line.
- We got a step closer with the discovery that √2 is an important factor embedded in all our Real numbers.
- We also know that the line of y=x corresponds to what we see in the prime counting function, π(x).
- Lastly, we know that prime numbers cooccur at the same rate along y=x.
We already have algorithms that will be able to quickly verify what we have so far. However, we still have a bit of an ambiguous indexing problem. It might be a little strange to have √2 as part of an index but it is probably doable despite aesthetics. What I want to know is what exactly about any number system would make them listable, indexable, or not and if that will be a problem we will need to address?
Just wanted to add this resource for anyone wanting to dig into number theory. NJ Wildberger is a Pure Mathematician and Professor of Mathematics at the University of New South Wales in Sydney Australia.
We will get to it soon but I’ll have to save what’s next for Part 2 to keep everything manageable. Thanks!