Saturday, May 23, 2009


JSH: Negative Pell's Equation existence conditions

The existence conditions for non-zero integer solution to j^2 - Dk^2 = -1, are:
  1. For every odd prime factor p of D, p = 1 mod 4.

  2. If D-1 has 8 as a factor it must have 16 as a factor as well.

  3. D cannot equal 1 mod 5.

  4. If D is divisible by 2 it cannot be divisible by 4.
Those are the first set of conditions required for the existence of the negative Pell's Equation.

There may be more, but probably not.

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