May 23, 2017
Hello readership! This is my first post in what I am hoping will end up being a fun and interesting blog. The inspiration for it comes from my recent (awful) experience in trying to get a math paper I wrote published in a math journal. The paper involved prime numbers and proofs of Legendre’s conjecture and Brocard’s conjecture. If you’re curious about the full history of these 100+ year old conjectures, feel free to look them up on Wikipedia. To summarize though, Legendre’s conjecture states that if you have two consecutive natural numbers, n and n+1, then there is at least one prime number located between the squares of those two consecutive natural numbers. Brocard’s conjecture is similar, and states that if you have two consecutive prime numbers, then there are at least four prime numbers located between the squares of those two consecutive prime numbers.
Both of these conjectures are pretty simple to state, so you would think they would have been proven by now. In fact, I’m pretty sure I HAVE proven them, and the proofs were fairly straightforward (though maybe a little difficult to explain). They just had to make use of a method that I’m calling Accommodation Theory (in case you’re wondering, yes, I partly chose the phrasing just because it sounded important).
So what’s the issue? Well, I guess when a math problem goes unsolved for 100+ years, people assume the solution must be incredibly complex, so they are immediately suspicious of anything that looks simple. They also assume that the solution should come from someone who in the least is trained in graduate level mathematics or beyond. My paper violated both of those rules. It offered proofs of two 100+ year old math problems in under 10 pages, and it was written by me, an individual who is not a math PhD or grad student, but a lawyer who happens to dabble in math in his spare time. As such, my work was deemed ‘untouchable,’ not even revisable, by the math journals I submitted to (I’ll refrain from naming their names).
Their comments were fairly brutal. Two journals rejected my work solely because I was not affiliated with any math department. One basically said they didn’t want to take the risk of publishing it (isn’t that the whole point of publishing academic pieces, to stir discussion and debate?). The final noted that my “enthusiasm is commendable,” but that the paper is not appropriate because my definitions were “cumbersome,” the introduction did not “clearly highlight” and summarize what was being done, and finally a good explanation was not provided for why I was successful “where so many others have failed” (I mean, I guess when you read the paper you can see that these are true criticisms, but (a) aren’t these revisable matters? (b) why do I need to explain why others have failed? and most importantly (c) I don’t see anything in there saying my work is incorrect!). After all of those rejections, I finally decided that I would submit my paper on an academic open source site called arXiv, but when I made an account and prepared to submit, something popped up stating that I needed at least one academic in the subject area to vouch for me before I could make submissions. If my whole difficulty so far was in getting through to academics in the subject area, how was I going to get one to vouch for me? Oi vey! At that point I was tired of academia.
Then it hit me. I really didn’t care about getting published or seeking glory in mathematics. I just wanted to share this cool thing I was doing called Accommodation with other mathematicians, and to see whether or not those mathematicians agreed that this Accommodation thing works to prove Legendre’s conjecture and Brocard’s conjecture (again, I’m pretty certain it works). Who needs a journal for that? All I need is a soapbox. A blog. So that’s exactly what I created here! A place where I could talk about the exciting math and philosophy that interests me, others can talk about the exciting math and philosophy that interests them, and it could all be done in a casual setting that doesn’t require some academic’s stamp of approval. Ahhh. Liberation!
So here it is. Attached below you can find my draft paper, with all its warts and potential insights on prime numbers. Read it, understand it, and then feel free to question or comment with your two cents in the comment section. Who knows, maybe by providing an organized open forum to discuss math concepts, we can push the subject forward in new and exciting ways. Or, in the least, we can just take some time to enjoy a subject area that always challenges.