Pell's Equation Rediscovered

One interesting equation I decided to study (for a reason I can't remember) was 2x^2+1=y^2, for positive integers x and y. I came across some very interesting conclusions in the relationships between x and y and patterns as the solutions grow larger and larger.

First, however, I would like to note that this equation has already been studied, quite extensively, by quite a few mathemetitions. It is known as Pell's Equation, and in a more general form is written as y^2-Nx^2 = 1. Traditionally it has been studied by looking at the smallest solution for x and y given a specific N. I approached it in a different manner, and found some relationships they seem to have missed. (I also eventually generalized the problem to Nx^2+1 = y^2 after getting bored with just N=2. I also tried out Nx^2 + m = y^2, given the flexibility that C++ allows.)

There were a few provable relationships. First, I noticed that as the solutions for x and y got larger and larger, there seemed to be some constant ratio between the two. (I considered y/x.) By approximation on my calculator, It seemed to approach the square root of 2. Fascinated, I realized that one could solve the given equation for y, yielding y = Sqrt[2x^2+1]. Then, all I had to do was take the limit as x goes to infinity of y/x, or

Lim[x-->infinity] (Sqrt[2x^2+1]/x)

Which is easily simplified to the square root of two.

Another strange set of relationships I noticed was between consecutive solutions. For example, consider one solution to the equation in (x,y); call it (a,b). Then, the solution immeadietly following that one is (c,d). As it turns out, a+b = d-c. I found this a striking result, and still I can't think of a way to relate a given solution to the solutions before and after it.

Determined to find something useful, I pressed on and came across some extremely far-fetched patterns I never would have suspected. It starts out simple. For each solution (x,y), x+y can be written as the sum of two square numbers. (I got excited, expecting some connection to Richey numbers; however, while some of these numbers can in fact be written as the sum of two squares in multiple ways, I haven't yet established a notable connection.) Then, those numbers began appearing all over the page- in old solutions! The relationship is quite difficult to explain, (in fact there are two), so I'll write out the solutions I looked at to formulate it.

(2,3) => Sum = 5 = 1^2 + 2^2
(12,17) => Sum = 29 = 2^2 + 5^2
(70,99) => Sum = 169 = 5^2 + 12^2
(408,577) => Sum = 985 = 12^2 + 29^2
(2378,3363) => Sum = 5741 = 29^2 + 70^2
(13860,19601) => Sum = 33461 = 70^2 + 169^2
...

I've bolded the important numbers, and it isn't difficult to see where the squared terms that compose the sum come from. The first, easier relationship is this: one of those terms comes from the sum of the solution before it. (I've written it in a way where you can see this: the 2 comes down from the first solution, the 5 comes down from the second solution, and so on.) However, you may notice where the "new" numbers are coming from. An easy way to put it is that the sum is re-written as u^2+v^2 in the (2n)th solution, where u is the x of the nth solution and v is the sum x+y of the nth solution. Also, the intermediate step (the 2n-1st solution) takes u from the sum of the nth solution and v from the x term of the n+1st solution. Though it may be difficult to sort out, the relationship is there.

There is more! Another, yet odder pattern I noticed is one concerning only the x part of the solutions. I saw that there seemed to be some common ratio of the x of the nth solution and the x of the n-1st solution, that Lim[x-->](Xn/Xn-1) == Some quantifiable value. Using calculators and some intuition, I found that this ratio (for N=2) is 3+2*Sqrt[2]. Furthermore, some ratio appears in the sequence of x solutions for all equations with a given N- for N=3 the ratio was Sqrt[3]+2; for N=5 it was 8*Sqrt[5], and so on. (N cannot equal a square number, because when it does there are no solutions to Nx^2+1 = y^2. I have not yet proven this result, but someone has (according to Mathworld) and I do think it is interesting.) Finally, these ratios do have somthing of a common pattern. Each one, given a particular N, contains the square root of N in it; it appears in the form a + b*Sqrt[N] for integral a and b. (Also, as far as I've noticed, a and b happen to be positive.)

Another part, which I haven't looked at in depth just yet, is something mentioned on wolfram's mathworld site. It says there that the solutions to the equation can in some way be quantified by continued fractions. I don't know much about continued fractions, and I plan on looking into it to see if anything can help me out.

(Check out http://mathworld.wolfram.com/PellEquation.html.)