Counting quadratic residue pairs

[JSH]

I noted that I figured out the reason for the size of solutions to x^² - Dy^² = 1, but also I found a way to count these things called quadratic residue pairs.  And if you've never heard of them, I hadn't either before I did it. After I noticed this thing I found out about them.

The thing is, you can add y^² to both sides:

x^² + y^² - Dy^² = y^² + 1, to group as: x^² + (1 - D)y^² = y^² + 1, which is:

x^² - (D - 1)y^² = y^² + 1 

The purpose of that is to move to something called modular arithmetic, so that I could get:

x^² = y^² + 1 mod D-1

And that forces these things called quadratic residue pairs.  And to prove things about them, I actually used a slightly different form:

(z-y)^² = y^² + 1 mod D-1 

Residues are just numbers less than the modulus, like mod 7, the residues are: 1, 2, 3, 4, 5, 6

And the quadratic residues come from squaring those mod 7, which means everything less than 7.  Like 6*6 = 36, but 36 mod 7 = 1. 

You see, with modular arithmetic, nothing can be greater than the modulus.

If you square all the residues for 7, you find there are only 3 quadratic residues and one quadratic residue pair:

1, 2, 4

Where I highlighted the quadratic residue pair, so it's quite simple: a quadratic residue pair is when one quadratic residue is 1 away from another. 

If you want to see more, here's another example mod 11:

1, 3, 4, 5, 9

And I figured out a way to count quadratic residue pairs, where it turned out I used two equations, including the one shown above:

2y = z - z^^-1 mod D-1 

(z-y)^² = y^² + 1 mod D-1 

And they give a new way to prove the equations for the count.  It's actually fascinating because it shows this one ancient math equation amazingly enough gets related to behavior of quadratic residues, and while math people teach ways to count quadratic residue pairs for primes, my way allows you to count for composites as well.

Want to learn more about how my research lets you count quadratic residue pairs?  Then go to my math blog:

http://somemath.blogspot.com/2011/10/counting-quadratic-residue-pairs.html 

James Harris

[Michael Barry]

I have done something similar here: https://www.mersenneforum.org/showthread.php?t=25164

[JSH]

Cool! Here is a place to post. As have explained do not review math from others but want to provide a place where someone may. Good luck.