Sunday, May 17, 2009


JSH: Negative Pell's Equation, consequences

A telling discussion erupted when I challenged these newsgroups on a simple result about the negative Pell's Equation and its connection to Pell's Equation, where posters claimed the result was previously widely known.

That result is that given j^2 - Dk^2 = -1, the negative Pell's Equation, x, for x^2 - Dy^2 = 1, is given by x = 2j^2 + 1.

I've noted that is not stated in mainstream mathematical literature. Posters have argued with me—often insultingly—claiming it is.

In response to a request for citations giving the result, they gave, if anything, citations to equations that could LEAD to that result, if you knew where to look!!! All math is derived from prior equations, except the basic axioms.

Now here are some obvious consequences of that result to further test your credibility in believing it is already part of mainstream literature.

Given non-zero integer solutions to x^2 - Dy^2 = 1:
  1. x must be odd, when a solution to the negative Pell's Equation exists.

  2. x = -1 mod D is required when a solution to the negative Pell's Equation exists.
The issue is whether that is mainstream knowledge in MODERN number theory, so whether or not you think you know number theory and know these trivial results is an issue because I am trying to teach here.

The people arguing with me are, in my opinion, hoodlums disrupting a class discussion.

Knowledge just is. But when some people hate knowledge because they think it hurts their class positions or their personal views of themselves, they can behave very badly in their attempts to hide knowledge.

I think Pell's Equation is kind of fun. There are interesting little results around it easily available to people who enjoy mathematics.

It's not my fault if these results are not taught by mainstream mathematicians.

And I should not be attacked for giving them now.

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