perplexus.info :: Geometry : Integer points on circles.

Integer points on circles. (Posted on 2011-02-28)

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
Puzzle Answer	K Sengupta	2023-07-31 00:28:39
re: OEIS sequence, NOT THIS ONE	Ady TZIDON	2011-03-01 16:37:47
OEIS sequence	Brian Smith	2011-03-01 11:26:35
further and further ...	Ady TZIDON	2011-03-01 11:20:45
re(2): n = 2	Steve Herman	2011-03-01 11:08:26
re: n = 3 (spoiler), difficulty d2?	Ady TZIDON	2011-03-01 09:11:08
re: n = 2	Ady TZIDON	2011-03-01 09:09:29
n = 4 + full solution (spoiler)	Steve Herman	2011-03-01 03:26:31
n = 5 (spoiler)	Steve Herman	2011-03-01 03:07:21
N=5	Brian Smith	2011-02-28 21:32:14
A symmetry argument	Steve Herman	2011-02-28 18:03:38
n = 3 (spoiler), difficulty d2?	Steve Herman	2011-02-28 17:57:40
n = 2	Steve Herman	2011-02-28 17:50:01
re: 1st four --d1	Steve Herman	2011-02-28 17:48:17
1st four --d1	Ady TZIDON	2011-02-28 13:50:12

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