Re: question: integer points on curves
- From: David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx>
- Date: Sat, 07 May 2005 08:18:22 -0500
On Sat, 07 May 2005 04:13:07 GMT, Nathan <beethoven3322@xxxxxxxxxxx>
wrote:
>Hi,
>
>I have a question about integer points on circles. Let R(N) denote the
>number of points in the plane with integer coordinates on the circle
>centered at the origin of radius N. Then I want to show lim R(N)/N -> 0
>as N->Infinity, but I've had some trouble showing it. I checked Niven
>and Zuckerman's "Theory of Numbers", but they give a formula for the
>number of solutions to x^2+y^2=n in terms of exponents of prime factors
>of n of the form 4k+1, and this doesn't seem to be very helpful. Any
>suggestions?
What _is_ that formula? Maybe it actually is helpful.
(My first impulse was to say that if you draw a picture and
think about it you can show that R(N) is <= the area of the annulus
N - 1 < sqrt(x^2+y^2) < N + 1.
But that only shows R(N)/N is bounded...)
>-Nathan
************************
David C. Ullrich
.
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