Note from the author#
Who am I? Doesn’t matter.
Why bother deeping primes like this? No particular reason except I love numbers (and don’t particularly like counting).
Why write a book on primes? Two reasons:
So that people can learn about productive numbers and do something cool with them.
To demonstrate how much fun mathematical research is in an accesible way.
What’s the point of this section? To go on a rant about mathematical snobbery.
When I first started this journey, many years ago, I knew very little math. I literally just thought modular arithmetic was cool and wanted to give multiplication some space to shine outside the shadow of the sucessor function. If I had known about the divisor lattice I probably would have stuck with that. But I didn’t so I followed my gut and somehow found something new (at least I’m pretty sure its new - very possibly this is known by some niche crowds who haven’t given it enough PR in which case shame on them).
At this point, I know a bit more math. I knew it was time to get make this public when everything started feeling obvious but I had no fresh ideas for next steps. I expect that most professional mathematicians would consider all the results so far trivial. To be fair, what took me several years would probably take an average grad student an afternoon to work out.
But I’m not ashamed of that. In my opinion, part of the beauty of math is the sheer simplicity of it. Not simple in the sense of what you most expect initially - our expectations are often clouded by previous experiences. Instead, I mean simple as a sort of self-contained completeness where, at least with hindsight, every “decision” follows the last in a way that’s so natural its almost inevitable.
I’m not an expert but I believe professional mathematicians are perversely incentivized to make their work appear as complicated as possible. Academic mathematics is this kind of elite club where everyone is constantly trying to prove smart they are. They fight over jobs, titles, prizes and whose name goes on the latest type of vector space or whatever. To survive they compete to publish in the fanciest journals, which requires the fanciest titles and makes the most concrete progress towards the open problems in their tiny niche specialism. All of this ego continues to feed the idea that math is only for people with special brains. So they forget the best math is painfully simple.
Ironically, amongst all this ego, mathematicians have a strange practice of refusing to acknowledge their own existence in their writing. I once read a paper with a single author which cited another paper by the same author saying “we refer the reader to our work”. What? Who is this imaginary we? Even worse is the pronoun-less “it is now shown that …”! One justification for all of this ‘we’ is that a math paper should be read like a collaborative journey between the author and the reader. I think that’s really cute and tried to embody it in this book. But that doesn’t explain why everything has to sound so objective and dispassoniate. If its a journey, why not make it an honest one? Why say “it is natural to assume …” rather than “the only thing I could get to work was …”? Every mathematician I’ve ever spoken to is in it for the beauty, so why does the whole section of their paper dedicated to motivation never mention this?
This difference in writing style goes all the way back to some dead Greeks: Euclid (who wrote like a robot and maybe didn’t even exist) versus Archmides (who interspersed proof with poetry and is famous for getting so excited about a geometry problem he jumped out of the bath and ran through the streets naked). Sadly, Euclid’s legacy has prevailed. This book has been a small little attempt to bring back bath math. With rhymes like that, you’ll be glad it didn’t include any poetry.
The reason I keep emphasizing the mortality of former mathematicians is because another thing I am sick of is genius worship. Of course, I am deeply grateful to the millenia of mathematicians who came before us and built the field into the glorious subject it is today. But the way we venerate the giants of old is stunting the progress of the future. There’s a popular youtube channel that reads papers by dead nerds and treats them with the veneration of holy scripture. I have spoken to mathematicians far more gifted than me who are afraid to try anything radical because they believe they are not suitable for coming up with new ideas. I believe these are linked. Do you think Euler (who is also dead) would have been inspired to make so many contributions if he’d been told his whole life that his textbooks were written by demigods?
To sum up:
Prods are simple and that’s a good thing.
I write with feeling because that’s how I feel, and hope you do too.
It doesn’t take a genius to find new math. From what I’ve gathered it takes three things:
Curiosity: to want to comprehensively understand, just for its own sake, whatever it is you’re interested in. With enough curiosity, you stop thinking “I could never understand this” and start thinking “I must understand this”.
Persistance: things rarely work the first time. You’ll make so many stupid mistakes. But who cares? Each time, you learn.
An internet connection: wikipedia, youtube and stackexchange are a pretty lethal combination. Chatbots will probably be a game changer soon too.
If you’ve got this far, you clearly have all three. So what are you waiting for?