So I was looking for an old video in numberphile (one about a particular kind of prime) and re-saw this
https://www.youtube.com/watch?v=yGsIw8LHXM8 (two square theorem)
or if you like the original video that one comes from
https://www.youtube.com/watch?v=SyJlRUBoVp0 The Prime Problem with a One Sentence Proof )
Anyway I found myself rewatching it. and decide to see if I can make a simpler version of the proof. not less space mind you, simpler as in more straight forward. I think I've done that here (it's probably not new, most things in math aren't) but regardless I submit it for your entertainment/utility.
Also it's worth saying that the you can then figure out exactly what form the X and Y (K or what have you) need to have by just unraveling all that. Here is that little bit of extra formula stuffs spelled out.
I feel like I've started many things and then abruptly stopped before i really got in to the thick of it. I'd like to share some of the highlights. Doing so will give anyone reading a good picture of where everything is with the stuff i normally (its been over two months) blog about.
Data Mining: Almost 6 months ago I was working on a GBM in a GBM model and there it has sat since then. It's not that I think it's merit-less, it is just that I doubt it'll produce amazing results. So, I'm not really inspired to finish it. Also, I don't have a kaggle contest to work on right now and that helps drive my interest in the algorithm. Truth be told, I'm beginning to think data mining is getting near the end of its "big gains" period. the human analysis part may very well be improvable but that's never interested me much.
Running: I started running, then i stopped, then I started again. to expand on this, it was fine, then i over did it, then I got motivated and eased back in. I've got a marathon in mind I want to go to. It is still 11 weeks away so I'm training up for that. I'll talk more about it when I commit to it fully (basically in about 4 weeks).
Math: I spent a lot of time trying to solve the 3 cubes problem. My most recent attempt sent me down a rabbit hole of general factoring. I actually thought I had a method for doing that, only to realize that my solution to the problem was such a tiny corner case as to be unusable. The net-net is I got nothing I'm pursuing here right now.
Magic: Things continue to wind down. there is a grand prix tourney in Dallas in 2 weeks but I don't think I'm gonna go. I still have fun playing most weeks but I'm definitely not feeling the drive to brew decks like i did. And, lets be honest, i've never felt the competitive spirit this time around. That is to say, I want to win sure, but deck building always took precedence. as it is so much more interesting than playing a known decent deck, which just makes it hard to be competitive.
EM Drive: Have you seen this thing? Its about 2 years old from a "oh hey something new!" perspective. but the last few days I've been reading up on the science and watching youtubes about it. I have to admit I'd like to better understand how it works. I totally get what virtual particles are, but I don't get how they can get the transfer of momentum to them (they are incredibly hard to interact with). Just something I've been messing with and thought I'd include as a bullet point. I doubt I'll build anything in my garage but crazier things have happened.
If you are reading this you might be searching for a solution, saw my comment or just be browsing the internet. So here's the thing, I saw a Numberphile and was like "i gotta try that" https://www.youtube.com/watch?v=Y30VF3cSIYQ . I do recreational math, so you know.... i like a challenge. anyway here's the solution I came up with (I circled) the "magic step". Basically the realization is that the left and right sides are in the same form. and since I made "I" up I can choose it to be equal to B which forces the other side to be A which then of course dictates its value.
This was my 2nd attempt I spent a few minutes going down another path to see if i could turn it in to a quadratic and just get solutions... it was a mess so I went back and looked for well, what you see. Oh and above The J substitution isn't necessary. i was just doing it to see if anything made sense as a next step beyond the obvious. it didn't help you could remove it. I just didnt want to rewrite the work and i did it in ink so... there it stands.
So I was watching/catching up on numberphile and watched this 74 is cracked - Numberphile . The short version is of all the numbers less than 100 they have shown that only 3 (now 2) can be expressed as the sum of 3 cubes except for a list of a few which are proven to have no solutions. I had watched the original video this one references before and while being entertained at the time, I promptly forgot about it after watching it.
I dont think many people realize or trully care that I've spent long years of my life thinking about Diophantine equations . I did it back when I was looking for new method to methodically factor primes (read 17-32 off and on) . While I came up with some interesting stuff (to me at least) I never really got anything that good. It was always a hobby/fun problem for me. The point is solving the 3 cubes problem is something the "work" i did could probably do relatively quickly if there arent tons of Residue Classes in the various modulos of 33 and 42 (the two remaining numbers). Just saying that makes me think there probably are, but if there arent merging them in to larger equations isn't very hard.
Even if there are bunches you can still get a new set of equations that dramatically cut down your search time. Getting past 10^16 is pretty remarkably easy heck getting past 10^32 might be trivial it just depends on the numbers. I think I'm gonna spend a little time on solving 33 and 42 just for fun and see how hard the problem really is. It would be neat to have my named tied to something in mathematics even if its just a simple solution to some rather mundane puzzle. And I need another hobby/would like to return to the old hobby for a while.