One of those things I like to say is--the math won't change. And that's true. Given a mathematical explanation, you have one forever. Well I have an odd situation though, where apparently a mathematical explanation doesn't appear to be popular with the people who supposedly care the most about such things--mathematicians.
So now there is maybe a unique in history chance to see what happens when experts in a field, and a very prestigious one, do not like a result, which they cannot touch, because you see, the math won't change.
Current accepted number theory about the equation x2 - Dy2 = 1 cannot explain the size in general of the fundamental solution, which is the set of smallest non-zero integer values for x and y that will fit!
My own research proves that D is key, and if D is a prime or two times a prime, then solutions will be the largest, with exceptions, when D-1, D+1, D-2, or D+2 is a square, as x is forced to equal 1 or -1 mod D, especially if D-1 has small prime factors, and if 4 is also factor they will be even bigger still.
If we were to consider a rating of that result from a scale of 1 to 5 based on the behavior of mathematicians whom I've informed of it, I'd think we'd see a 1.
And note, mathematicians currently have NO explanation of their own for the behavior of this equation.
And maybe it'd help to see that behavior so I'll put up the solutions from 20 to 30.
I'm not calculating these myself but found them in an article on what is generally called "Pell's Equation" on a popular math site called MathWorld: mathworld.wolfram.com/PellEquation.html
I picked a range arbitrarily that shows I think a healthy variation of solutions. I'll give the equation with the solution and the explanation under the rules follows.
92 - 20(2)2 = 1
Starting, 20 is NOT prime or twice a prime, and 20-1 is 19, which is prime, and that tends to give small solutions.
552 - 21(12)2 = 1
Here D is not prime or twice a prime as well.
1972 - 22(42)2 = 1
Here D is twice a prime, and notice a slightly bigger solution. And D-1 is 21, two prime factors.
242 - 23(5)2 = 1
Now a new part of the rules shows itself, as D+2 is a square! So back to a small solution.
52 - 24(1)2 = 1
This time 24+1 = 25, so D+1 is a square, and you can see this pattern through 4.
No solution exists for D=25 because it's a square.
512 - 26(10)2 = 1
On the other side of 25, so still smaller solutions as now D-1 is a square.
262 - 27(5)2 = 1
And last easy one near a square as D-2 is a square.
1272 - 28(24)2 = 1
Here things surprise me a bit, as I think it's larger while D is not a prime or twice a prime, but maybe that is somewhat subjective.
98012 - 29(1820)2 = 1
But now is a much bigger one, where D is a prime, and D-1 has 4 as a factor.
112 - 20(2)2 = 1
And back to tiny with the last example, with D not a prime or twice a prime.
Remarkably, mathematicians believed the solutions bounced around with no explanation.
The actual explanation wasn't just me looking at a pattern either. As I went to see the pattern after figuring out mathematically what behavior should occur. Which is actually a lot of fun. It's like theory first, and then going to look and seeing behavior fit theory! Lovely.
So what is happening? Why this behavior?
The need to wonder 'why' is fundamental to human behavior. We are a curious species.
The answer is actually remarkably cool. The equation x2 - Dy2 = 1 actually generates curves. It will produce ellipses if D is negative, and with D positive, like with our integer solutions above, if you graph it, you will get hyperbolas. So I like to call it the two conics equation.
It turns out that the hyperbolas, connect to other ellipses, from a completely different equation! And that connection forces integer solutions to be larger. And only occurs if something special happens, which is if x = 1 mod D or -1 mod D, which just means, if x-1 has D as a factor, or x+1 has D as a factor. And that happens if D is a prime or twice a prime, though it can also occur in other cases as well.
Consider: (x+1)(x-1) = Dy2
For example with our biggest solution above with D=29, and x=9801. And x+1 has 29 as a factor, as:
9802 = 29*338
I show the mathematics, in a post: somemath.blogspot.com/2011/09/two-conics-equation-size.html
Now then, back to the issue of rating the math! It turns out that whether human beings like it or not, this behavior IS the 'why' of the behavior of the equation. It is possible to prove using mathematics which should be trivial to a trained mathematician.
Yet here I am explaining it again, when my post above was made by me on September 20, 2011.
I think you can validate that 1 star rating from the math people from that information, including the information that I've diligently informed them of this result, and pressured them to acknowledge it.
And the math will not change.
So where went human curiosity? Love of knowledge? Where is the mythical "mathematician" who would presumably relish an explanation where there was none?
Good questions. If people don't ask them, then we just won't know, as I'm not the person with the answers there!
It IS a fascinating situation in many ways, regardless. It lets us see in our celebrity obsessed world, what might happen if popular opinion doesn't like something that is true without concern about opinion.
It is a wonderful and unique opportunity.