The problem you pose is notoriously difficult, I
think.
I was actually thinking about another problem,
specifically, to show
that for any n >= 0, there exists
a circle whose interior includes
precisely n integer
points.
An outline of my proof:
Let O be a point in the plane which is not
equidistant from two
integer points. Then any circle centered at O
passes through at
most one integer point. Take the sequence
{C_n | n = 0, 1, 2...}
of circles centered at O and passing through an
integer point,
ordered by increasing radius.
We can show that C_n has
exactly
n points in its interior.
Now we need only show that O exists.
Certainly any O = (x, y)
with mutually transcendental coordinates works, but
I was hoping
to find point O with explicit coordinates.
This got me wondering
exactly which points
O were candidates, specifically, exactly which
points of the plane are not equidistant from two
integer points.
----- Original Message -----
Sent: Thursday, January 22, 2004 1:54 AM
Subject: Re: [math-fun] Simple Question I think
David Wilson asks:
<<
Is every rational point on
the plane equidistant from two integer points?
>>
Here's a
question in this vein that I think is unsolved:
In the plane,
given a unit square, is there any
point whose distance to each corner
is rational?
Dan
--
NOTE: Please direct any responses to
<asimov@msri.org>, since I may not
keep this AOL e-address much longer --
thanks.
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