Considering a behavior

One of the things that helped me a lot recently was a result of mine with a famous equation, where finally I could look at something that seemed to explain a lot for me. So for background information for those interested in my research and issues around it, I want to talk about this result as objectively as possible. So deliberately I'll try to stay away from emotion, and focus on the facts.

The famous equation is x² - Dy² = 1, which according to various web sources has been known for over a thousand years. Typically integer solutions for x and y, given positive integer D, are discussed in regard to it, for instance: 3² - 2(2)² = 1

However, the size relative to D of the smallest set of positive non-zero integers which will give a solution, which is called the fundamental solution, can vary extensively.

Current accepted number theory about the equation x² - Dy² = 1  cannot explain the size in general of the fundamental solution.

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.

That is, x-1 or x+1 must have D as a factor, if D is a prime or twice a prime, which forces a larger solution unless D-1, D+1, D-2 or D+2 is a square. And when a larger solution is forced, if D-1 has small prime factors, solutions will be even larger. And then if 4 is also a factor they are the largest.

For example there is a large fundamental solution at D = 61.

The smallest positive non-zero x and y that will work are x = 1766319049 and y = 226153980.

1766319049² - 61(226153980)² = 1

Currently accepted number theory cannot tell you why that solution is so large, as my research is to my knowledge not yet accepted.

Here notice that x = -1 mod 61, as: 1766319049 + 1 = (28956050)(61)

From my research it follows that the solution is so large because x = -1 mod 61, D-1, D+1, D+2, and D-2 are not square, while D-1 = 60, which has the first three primes as factors, as 4(3)5) = 60, and because 4 is a factor as well.

So I can explain why D=61 is so large and my explanation is backed by a mathematical proof which has been public on this blog since September 2011.

Also I directly contacted various mathematicians last year, giving them the rules. Usually I went to some effort to find number theorists so that the people contacted would presumably have an interest in this result.

Also one of the mathematicians I knew of from searches on the equation commonly called "Pell's Equation" by mathematicians as he had a book listed on Amazon.

Contacts were successful to some extent in that I got emailed replies from several mathematicians at first, but later, no additional emails garnered replies.

To my knowledge there has been no acknowledgement of this result by established mathematicians.

The equation is very famous and presumably is taught yearly in number theory courses.

As far as I know, mathematicians do not explain the size of the fundamental solutions at all.

Remarkably you can find papers on the subject, however, for instance:

"ON THE SIZE OF THE FUNDAMENTAL SOLUTION OF PELL
EQUATION" by ETIENNE FOUVRY.

It is a 29 page paper and I looked through it trying to find an example with an actual D. There was none. I also scanned through looking for an explanation for the fundamental solution size. I saw none, but I'm not a mathematician.

Another reference to a paper where I could only see the abstract:

"The size of the fundamental solutions of consecutive Pell equations" by Michael J. Jacobson and Hugh C. Williams

There the abstract indicates, well I'll just quote part of it:

"...We also provide some heuristic reasoning which suggests that there exists an infinitude of values of D for which $\rho(D) \gg \sqrt{D} \log \log D / \log D$, and that this is the best possible result under the Extended Riemann Hypothesis. Finally, we present some numerical evidence in support of this heuristic."

I could present more examples but I think those two are indicative.

Looking over what I can see on the web, modern mathematicians cannot explain the size of the fundamental solution to the equation x² - Dy² = 1.

But also, now a year since I informed some of them, they have yet to acknowledge the solution.

That is the behavior under consideration in this post for background.


James Harris