Correct solutions were submitted by Lou Cairoli, Agustin Murillo, Ross
Millikan.
Perhaps the simplest example is x2 +
2x
y + y2 - 2 = 0. This has no
rational points since the equation can be transformed to read (x + y)2
= 2, and any rational points would provide a rational square root of 2, which is
impossible.
For a fun set of examples, first pick a positive integer, n
> 2, then observe that any rational point solving the equation
xn + yn = 1
would provide a counterexample to Fermat's Last
Theorem!
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