The day I thought I discovered something
I'm a little boy who'll one day grow up to be a
wise yet not
benevolent man. To make a head start, I've been trying to figure out the simple yet hard
conjectures, such as
Goldbach's Conjecture or
Sierpinski's Conjecture. They're both to do with primes; I don't know why, I'm not especially attracted to
number theory. In fact, I don't even understand why number theor is such a fuss; what's so cool about integers? But so it is.
So, today (well, yesterday) I was again pondering on that damned problem. I realized that a highschooler like me is very unlikely to solve anything, but hey, you can always try, and it's certainly more interesting than studying swedish. In fact, I thought I had actually made progress, although it turned out to be nothing special after all. For those who don't want to click the link on goldbach's conjecture, I'll say that it simple states that for every even integer x, there are two primes p and q so that p+q=x. Anyways, I picked one integer, S=26, and drew a map of modulos to this with every integer smaller than S, so that on the line 2 of chart I have cross at columns zero, two, four, ... since mod(26,2)=0, at line 3 the cross is at two, five, eight, ... and so on.
Now, I had already figured, if map for S is like this, what does S-T look like? Simple, just move all the crosses t steps left. And so it was. Then I looked at it and knew that those lines that have cross in column zero are the factors of S. From this I knew that if I find a column T with one cross, then S-T must have only one cross in column 0, that cross being S-T's only factor - a prime! Now I knew that if there's only one cross in a column, then S-T must be a prime, and also, the line on which this cross was must be a prime too. I was quite excited about this; it meant that if I could prove that map for every S contained at least one prime column with only one cross, then this conjecture would be proven.
Today (yesterday), I was even more excited when the seemingly random image of crosses suddenly gained form. I realized that it could be drawn geometrically, by drawing lines from the coordinates (s,0) to lower left with angle ratios 1/i, where i is every natural number larger than zero. I contemplated on this, and realized... that I had gone back to starting location. It meant that columns with only one cross were the columns whose distance from the right border is a prime. So, I had to find column T where T is a prime, and distance from right border (S-T) is a prime... umm... back to the starting location, I suppose. Unhappy.
But wait, there more; I didn't give up, although I had succeeded at nothing. I took my TI-89 from the bag, wrote a script to find the primes P,Q for any S. I knew that it had been experimentally proven that there are such primes for any S up to [damn large number here]. But how many such pairs there were? So, I wrote the script to count the solutions and draw a graph. It was kind of slow, but I had a whole day, so I let it do its job and studied modernism in literature while the calculator cranked up numbers and plotted them into a graph. The graph turned out to be quite a mess, a cloud of seemingly random dots, but there was shape. And sure enough, when I drew a 10-number moving average, it was apparent that there was shape! This was even more exciting than the previous one; you see, the shape looked very familiar. I remembered once hearing that there are about x/log x primes smaller than x. So, I tried if there would be by average x/log x P,Q-pairs for number x. No, there wasn't, but x/(log x)^2 was indeed very close!
As the rest of the schoolday was over, I went to batmud and asked a friend studying mathematics at university of helsinki. Once I managed to explain it understandably, he agreed that it was interesting. On the bus trip home, I verified that this estimation holds decently for mighty 2*10^3 (2000, but it looks larger that way ;) ) that my TI-89 could check. Exciting! If for some even number x there are about 1/log x primes smaller than x and (1/log x)^2 prime-pairs that sum up to x, then the ratio of the number of these pairs to primes is the same as ratio of primes to integers.
Of course, I was then all the more disappointed when I went looking for optimized prime finder program and instead discovered that someone had figured this, and much more, out a long time ago.
Well, it was fun while it lasted... back to playing StarCraft I guess. Otherwise, today (yesterday) was rather typical day. I liked my horoscope in ilta-sanomat that day, but since I don't believe in horoscopes it won't work. Sad, I really would have wanted it to work.