Counting Lattice Points Inside a Circle of Radius 2√2

Understanding the distribution of lattice points inside a circle is an intriguing topic in number theory and has implications in various fields. This article delves into the specific case of a circle with radius (2sqrt{2}) and center at the origin, providing a comprehensive analysis and demonstrating a method to count the lattice points within it.

Introduction

The concept of lattice points, points with integer coordinates on a plane, is fundamental in analytic number theory. In this context, we aim to determine how many lattice points lie strictly inside a circle of radius (2sqrt{2}) centered at the origin. This problem can be approached through both theoretical analysis and practical computation.

The Problem

Consider a circle with a radius of (2sqrt{2}) and centered at the origin (0, 0) on the coordinate plane. Our goal is to identify and count all the lattice points that lie strictly within this circle. A lattice point is a point ((x, y)) where both (x) and (y) are integers.

Understanding the Geometric Setting

A circle of radius (2sqrt{2}) centered at the origin can be described by the equation:

[text{Circle: } x^2 y^2 Our task is to find all integer solutions ((x, y)) to the above inequality.

Counting Lattice Points

To count the lattice points, we will systematically examine all integer solutions ((x, y)) that satisfy (x^2 y^2

Systematic Approach

We start by checking all possible integer values for (x) and (y) within the range that keeps the sum of their squares less than 8:

(x) and (y) can take values from (-2) to (2) since larger values would result in the sum exceeding 8.

We will list all valid ((x, y)) pairs and then count them.

Listing Lattice Points

The following table lists all valid lattice points for ((x, y)) that satisfy (x^2 y^2 xy -2-2 -2-1 -20 -21 -22 -1-2 -1-1 -10 -11 -12 0-2 0-1 00 01 02 1-2 1-1 10 11 12 2-2 2-1 20 21 22

Counting the entries in the table, we find there are 21 lattice points that lie strictly inside the circle.

Verification with a Grid

To provide a visual confirmation, we can also draw a 4x4 grid centered at the origin. The corners of this grid will lie on the circle, verifying our numerical result. A 4x4 grid includes points with integer coordinates from (-2) to (2), aligning perfectly with our earlier findings.

Conclusion

By using a combination of theoretical analysis and practical enumeration, we determined that there are 21 lattice points that lie strictly inside the circle of radius (2sqrt{2}) centered at the origin. This result is a solid example of how lattice point problems can be approached and solved systematically.