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Integer points on circles. (Posted on 2011-02-28) Difficulty: 3 of 5
If a circle is centered on the origin and has radius r, it may pass through some points with integer coordinates. In fact, by symmetry, this will be a multiple of 4.

Find the smallest radius that will pass through 4n integer points where n=1,2,3,4,5.

Feel free to go further.

  Submitted by Jer    
Rating: 3.5000 (2 votes)
Solution: (Hide)
r=1 has 4. (1,0) etc.

r=√5 has 8. (2,1) (1,2) etc.

r=5 has 12. (5,0) (4,3) (3,4) etc.

r=√65 has 16. (8,1) (1,8) (7,4) (4,7) etc.

r=65 has 20. (65,0) (60,25) (25,60) (52,39) (39,52) etc.

See http://oeis.org/A071383 for more terms.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Solutionre: OEIS sequence, NOT THIS ONEAdy TZIDON2011-03-01 16:37:47
Hints/TipsOEIS sequenceBrian Smith2011-03-01 11:26:35
Solutionfurther and further ...Ady TZIDON2011-03-01 11:20:45
Some Thoughtsre(2): n = 2Steve Herman2011-03-01 11:08:26
Hints/Tipsre: n = 3 (spoiler), difficulty d2?Ady TZIDON2011-03-01 09:11:08
Hints/Tipsre: n = 2Ady TZIDON2011-03-01 09:09:29
Solutionn = 4 + full solution (spoiler)Steve Herman2011-03-01 03:26:31
Some Thoughtsn = 5 (spoiler)Steve Herman2011-03-01 03:07:21
Some ThoughtsN=5Brian Smith2011-02-28 21:32:14
Some ThoughtsA symmetry argumentSteve Herman2011-02-28 18:03:38
n = 3 (spoiler), difficulty d2?Steve Herman2011-02-28 17:57:40
n = 2Steve Herman2011-02-28 17:50:01
Some Thoughtsre: 1st four --d1Steve Herman2011-02-28 17:48:17
Some Thoughts1st four --d1Ady TZIDON2011-02-28 13:50:12
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