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Saturday, May 30, 2015

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
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