As hard as number theory can be there are areas where you can stumble across simple ideas which have HUGE consequences, especially for generating controversy. Oddly enough it's easy to find raging arguments over seemingly basic things, where I stumbled into one by speculating about prime residues.
And it's easy enough to talk about it using 3 and some other primes. For instance 5 mod 3 = 2, while 7 mod 3 = 1. And if you're a coder you may be more used to seeing something like 5 % 3 = 2, and 7 % 3 = 1. And I use that so it may look familiar as I use "mod" all over the place, but now you should have the idea.
Well, I'm like, why should 3 have any particular residue modulo a larger prime? And "modulo" is just the more precise way to say what I showed above. And it turns out that's not a new thing to ponder and don't worry, we're close to one of the biggest raging arguments you can face if you dare.
For instance 101 is prime, and 101 mod 3 = 2. Think anyone could predict that without checking? Let's go higher. And 1621 is prime and 1621 mod 3 = 1. And no, wasn't trying to get that as just randomly went up to a higher prime. But it is convenient as you can have 1 or 2 as the residue, and why should it matter?
Well accepted number theory is that there is some very complicated mathematics which yup, does involve the Riemann Hypothesis, which says it DOES matter and is not random. While I suggested it WAS random and ran into some of the hugest arguments I've ever faced.
So I'm like, why should primes care what their residue is modulo 3? And extrapolated to primes in general modulo a larger prime and came up with some rules, and even came up with a prime gap equation. Did that years ago and turns out I have things I worked on years ago lying around that I just think about again now and then. Found the prime gap equation is in a post on my math blog from 2006:
somemath.blogspot.com/2006/08/prime-gap-equation.html
If you're bored and like playing with numbers and don't care if it's using ideas supposedly incorrect, you can code it. Mainly just need to know what a prime is, and have a list of primes to plug into this probability equation. Hardest thing to figure out is the correction.
Some may wonder why is it so controversial and why would primes supposedly have a preference. Why would 3, you could say, care whether any particular prime has 1 or 2 as a residue?
Well if there is no mathematical reason then it's random.
Turns out you could define "random" using prime numbers. Which I've done, but yanked it as too controversial.
But it turns out you could solve some currently considered unsolved math problems like the Twin Primes Conjecture and the Goldbach Conjecture. Easy.
So yeah, no way it's accepted mathematics in this day and age! And in fact if I were a mathematician I'd probably not even dare bring it up for fear of destroying my career immediately. Turns out I'm not the first person to come up with these ideas. But I remember finding out about some controversial mathematician who put this notion forward years ago, and later when making a post about it, couldn't find his name again. Otherwise I'd give it here.
Later I decided that I would go even further and conclude I'd actually found a prime residue axiom.
An axiom is a really cool thing where most people believe they were all found long ago. It's something believed to be self-evident. That is, its truth is supposed to be so obvious as to be not in doubt.
Turns out mathematics is built on axioms. Without them, we wouldn't have mathematics! These are the foundation level things, where there is no proof, just the sense these things must be true.
somemath.blogspot.com/2010/02/prime-residue-axiom.html
And I decided this no preference of prime ideas was an axiom I guess in 2010 based on the date of the post.
Since I made that up, if you web search on prime residue axiom, it's about my ideas, including people trying to convince they are wrong!
So much fun! Here we started with this simple idea about the prime number 3, and its residues, considered with other primes, and I've managed to take you through a journey through some of the most controversial territory in all of number theory. Managed to put forward the idea that randomness itself can be defined using prime numbers. And let you in on the possibility of a simple resolution to supposedly open problems like Goldbach's Conjecture all in one little post.
And I remind you, officially from mathematicians, there is no acceptance of a prime residue axiom. There is no accepted resolution of the Twin Primes Conjecture or Goldbach's Conjecture. And randomness is not considered to be well-defined through prime numbers.
But if I'm right, and in the future what I say is accepted then these are some of the most important ideas of all time. Just defining randomness changes how we look at random processes, including, yup, the Big Bang itself.
And getting back to the beginning of the universe looks like a good place to stop.
To me this example is one of the most fascinating you can come across to show how easily controversy can be found and to raise questions about what happens as a result. Am I right about prime preference? Let's say for the sake of argument I'm not. But isn't it fascinating to play with such ideas regardless?
Some would say no. And I think here you can see how that kind of pressure can remove from others even the knowledge of a thing.
But if it turns out I'm right then that's huge! If I'm wrong, no big deal really, for me. So for others I think it important to emphasize I'm going against the official position of mathematicians. And I'm NOT a mathematician.
I came into these ideas out of curiosity and discovered I wasn't the first person. But even finding that out can be difficult as mathematicians seem to prefer such things not even be mentioned.
Luckily, I'm not a mathematician, which I like emphasizing, so I am willing to take you on a journey from a seemingly simple idea to the edges of human knowledge and into some of the most controversial ideas in mathematics itself.
I love talking about ideas! And sometimes it's fun to take you to the edge, and show you how close it is! For me, it's where I often have traveled, in journeys for me that feel like a long time ago.
Now, most of the time, I prefer to stay firmly on accepted ground.
James Harris
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