Lattice Points on Circle: Determining the Number of Points on the Boundary

In summary, the conversation discusses the existence of lattice points on the boundaries of circles with irrational radii. It is determined that not all circles have no lattice points on their boundaries, as circles with radii √2 and √5 do contain lattice points. The definition of lattice points is clarified as points with integer coordinates.
  • #1
funcalys
30
1
Does any circle having irrational radius have no lattice points on its boundary ?
Extended question: Is there any way to determine the number of lattice points lying on the boundary of a given circle ?
*The centres of these circles are all (0,0) *
 
Last edited:
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  • #2
What do you mean by lattice points? Points (x,y) where x and y are integers?

The circle with radius 1/sqrt(2) comes to my mind.
 
  • #3
Thanks, but the equation x^2 + y^2 =1/2 seems to have no integer solution...
 
  • #4
Isn't that what you asked for?
 
  • #5
Ah, my bad :-p, I meant to ask if EVERY circle having irrational radius have no lattice points on its boundary, not an example :smile:.
 
  • #6
The boundaries of many circles having an irrational radius contain lattices points. For example, can you find lattice points on a circle of radius √2? What about one with radius √5?
 
  • #7
Petek said:
The boundaries of many circles having an irrational radius contain lattices points. For example, can you find lattice points on a circle of radius √2? What about one with radius √5?
Thanks, I didn't think thoroughly before posting this silly question, sorry.
 

FAQ: Lattice Points on Circle: Determining the Number of Points on the Boundary

What are lattice points on a circle?

Lattice points on a circle are the points where the coordinates are both integers on a Cartesian plane with the center of the circle being at the origin.

How many lattice points are there on a circle?

The number of lattice points on a circle depends on the radius of the circle. If the radius is an integer, there will be 4 times the radius number of lattice points. If the radius is a fraction, there will be 4 times the numerator of the fraction of lattice points.

What is the equation for finding lattice points on a circle?

The equation for finding lattice points on a circle is (x-a)^2 + (y-b)^2 = r^2, where (a,b) is the center of the circle and r is the radius.

How can lattice points on a circle be used in scientific research?

Lattice points on a circle can be used in various scientific research, such as in crystallography, where they represent the positions of atoms in a crystal lattice. They can also be used in computer graphics and simulations to generate circular patterns.

Are there any real-life applications of lattice points on a circle?

Yes, lattice points on a circle have real-life applications in fields such as optics, where they represent the positions of diffraction grating lines, and in signal processing, where they are used to generate circular patterns for antenna arrays. They can also be used in navigation systems to calculate the distance and direction from a certain point on a circle.

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