Can you explain this mathematical absolute?

Here is a fun little result, which is a mathematical absolute, with all positive integers:

x + y + j + 1 = n² or 2n²

if x = 1 mod D, x² - Dy² = 1, and j = (x+Dy-1)/D

For example, 8² - 7*3² = 1, so x = 8, and 8 mod 7 = 1.

j = (8 + 7*3 - 1)/7 = (7 + 7*3)/7 = 4

8 + 3 + 4 + 1 = 16 = 4²

That's a simple example to make this post easy to write for me, but the result is true over infinity.

So whenever x = 1 mod D, with x² - Dy² = 1, then these rules are forced, absolutely.

Seem easy or trivial?

It's another key piece of the explanation for an ancient math mystery.


James Harris