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From: OsherD on 4 May 2010 02:31
From Osher Doctorow

Graham Everest and Thomas Ward of U. East Anglia Norwich U.K., in "The
repulsion motif in diophantine equations," arXiv: 1005.0315 v1
[math.NT] 3 May 2010, 17 pages, relate number theory including
diophantine equations throughout their paper to repulsion.

The general idea is something like this:

1) A theorem about an equation having only finitely many integral
solutions is equivalent to saying that the point at infinity repels
integer points or that there is a punctured neighborhood of the point
at infinity that has no integer points.

They begin with the elementary proof that (3, +/- 5) are the only
integral solutions to y^2 = x^3 - 2, which involves unique
factorization:

2) (y + sqrt(2)i)(y - sqrt(2)i) = x^3

This is then generalized to a theorem that replaces -2 in the equation
to -d, d a nonzero integer, using a similar unique factorizataion.

Readers may notice that y + sqrt(2)i is similar in form to 1 +
sqrt(3)i and x + sqrt(3)iy, etc., from the last few posts, and that
the complex conjugate occurs in both scenarios.

Osher Doctorow

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