Earlier today, I talked about my fascination with square numbers, and their properties. Now I’m going to take it a step further and see what else we can do with square numbers.
When I was first introduced to geometry, I was immediately fascinated by the Pythagoran Theorem. This is the formula for finding the hypotenuse of a right triangle.
(Again, I’m sticking most of this in the extended section. And here’s fair warning: I have never seen any of this explained anywhere else, so in many cases I’ve had to invent my own terms and language to describe what I’m thinking about. If anyone gets confused trying to follow the way my brain works (something that happens to me on a fairly regular basis), feel free to ask me to explain better what the hell I’m talking about. It might take me a little time to answer it, but I WILL get to you.)
For those of you unfamiliar with it, let me explain: if you have a triangle where two of the sides meet at a 90-degree angle, you can use the two shorter sides to determine the length of the third side. If A and B are the sides that meet at the right angle, then let C be the length of the third side. Simply square A and B, add them together, and take the square root of that number. Or,
A^2 + B^2 = C^2.
The simplest example is 3/4/5. Simply put, 3^2 + 4^2 = 5^2. Or, 9 + 16 = 25.
I was obsessed with this, and I started looking for more sets where all the numbers were whole numbers, not fractions or decimals. And they started popping up with almost frightening regularity. 5/12/13. 8/15/17. 7/24/25. 20/21/29. 12/35/37. 9/40/41. And so on. And so on.
But amidst the seeming randomness, I started to discern patterns.
The first one is that there is a simple way to determine whether a number can be used for C. All it has to do is fit two very simple criteria:
1) It has to be a prime number
2) When divided by 4, it has to have a remainder of .25
If it fits both rules, then it qualifies, and determining the values of A and B is simple process of elimination.
(This only works for “pure” triples, when the numbers have been simplified. I don’t count 6/8/10, because it’s just the 3/4/5 triple multiplied by 2. I have limited myself to prime numbers for C.)
Then I started getting “cute.” What happens when you start multiplying two C’s together? what sort of odd things might happen then?
Strange things. You start getting multiple solutions — usually 4 different sets of A and B.
For example, let’s start with a simple one. 5 and 13 are the first two Cs, so let’s multiply them together and see what sort of As and Bs we get from 65.
The first two are obvious: 39/52/65 and 35/60/65 are simply 3/4/5 and 5/12/13, multiplied by 13 and 5 respectively. But there are two more solutions that work: 16/63/65, and 33/56/65. (I won’t go into the details of how I found them, as that gets VERY strange, but feel free to test them yourself and see I’m not cheating.)
I tried it with a few more simple ones, and I started to notice a pattern: the “new” solutions always involved a multiple of the number 7. In the above example, 56 and 63 are both divisible by 7. What the hell is going on here? Where did those 7s come from?
I thought I had uncovered something very strange, almost mystical here. But I soon found that the rule was not an absolute. When I tried 5*29, or 145, the “new” solutions (24/143/145 and 17/144/145) didn’t yield any new multiples of 7.
But the “old” solutions did: 20/21/29, when multiplied by 5, yields 100/105/145, and 105 is simply 7*15.
So what was the rule at play here? It seemed simple: If both or neither C had a multiple of 7 as one of its As or Bs, then the “new” solutions would have factors of 7. But if only one of them did, neither of the new solutions would. I tried it out, empirically, and it seemed to hold true.
The final step I’ve taken is in determining whether a C will have any 7’s in its triples. There had to be some sort of universal rule for determining it, some simple test to devise that would allow me to know whether or not it would, without having to calculate the solution. And through a lot of trial and error, I found the answer:
If a prime number can be expressed as a multiple of 7 PLUS a factor of 2, then it will have a 7 in its solution. If it can only be expressed as a multiple of 7 MINUS a factor of 2, then it will not.
I tried it out on a bunch of numbers, and it seemed to hold up. 29 is 21+8, so it’s good. 37 is 21+16, so it works, too. 41, though, is 42 – 1, so it doesn’t. And so on. And so on.
What is the next step? I have no earthly idea. I don’t know where I’ll go from here, if anywhere. And I don’t have enough formal math education to know if I’m on to something wholly new, something discovered centuries past, or am simply making stuff up. Part of the reason I’m tossing this out is to see if it seems familiar to anyone, who might point me towards actual scholarly work in this area. Another reason is I’m wondering if anyone else has been playing in this area, and shares my obsession.
And part of the reason is to show folks that no matter how strange you think my political opinions are, they’re nothing compared to what my mathematical ideas can be.