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Lattice points on a circle.

  1. Nov 4, 2012 #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: Nov 4, 2012
  2. jcsd
  3. Nov 4, 2012 #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.
  4. Nov 4, 2012 #3
    Thanks, but the equation x^2 + y^2 =1/2 seems to have no integer solution...
  5. Nov 4, 2012 #4


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    Isn't that what you asked for?
  6. Nov 4, 2012 #5
    Ah, my bad :tongue:, I meant to ask if EVERY circle having irrational radius have no lattice points on its boundary, not an example :smile:.
  7. Nov 4, 2012 #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?
  8. Nov 4, 2012 #7
    Thanks, I didn't think thoroughly before posting this silly question, sorry.
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