4. A lattice point is a point (ax, y) in the plane, both of whose coordinates are integers. It is easy to see that every lattice point can be surrounded by a small circle which excludes all other lattice points from its interior. It is not much harder to see that it is possible to draw a circle which has exactly two lattice points in its interior, or exactly 3, or exactly 4, as shown in the picture below. :O: :: ● Do you think that for every positive integer n there is a circle in the plane which contains exactly n lattice points in its interior? Justify your answer.
4. A lattice point is a point (ax, y) in the plane, both of whose coordinates are integers. It is easy to see that every lattice point can be surrounded by a small circle which excludes all other lattice points from its interior. It is not much harder to see that it is possible to draw a circle which has exactly two lattice points in its interior, or exactly 3, or exactly 4, as shown in the picture below. :O: :: ● Do you think that for every positive integer n there is a circle in the plane which contains exactly n lattice points in its interior? Justify your answer.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 74E
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Please do not rely too much on chatgpt, because its answer may be wrong. Please consider it carefully and give your own answer. You can borrow ideas from gpt, but please do not believe its answer.Very very grateful!Please do not rely too much on chatgpt, because its answer may be wrong. Please consider it carefully and give your own answer. You can borrow ideas from gpt, but please do not believe its answer.Very very grateful!
Transcribed Image Text:4. A lattice point is a point (x, y) in the plane, both of whose coordinates are integers. It
is easy to see that every lattice point can be surrounded by a small circle which excludes
all other lattice points from its interior. It is not much harder to see that it is possible to
draw a circle which has exactly two lattice points in its interior, or exactly 3, or exactly
4, as shown in the picture below.
:O:
●
●
●
Do you think that for every positive integer n there is a circle in the plane which contains
exactly n lattice points in its interior? Justify your answer.
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