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19-Tanzanite

## All distances of the points on the sphere to be Integer

All distances of the points on the sphere to be Integer

n=2, r=0.5, l=1 (0.5,0,0),(-0,5,0,0)

n=3,

10 REPLIES 10
19-Tanzanite
(To:ttokoro)

n=4

19-Tanzanite
(To:ttokoro)

n=4 of 2D;

The one of 3D;

n=5 of 2D; by terryhendicott

The one of 3D;

19-Tanzanite
(To:ttokoro)

n=5 with new 2D n=4 results.

19-Tanzanite
(To:ttokoro)

n=6, r2D6p=

19-Tanzanite
(To:ttokoro)

19-Tanzanite
(To:ttokoro)

19-Tanzanite
(To:ttokoro)

2-Guest
(To:ttokoro)

@ttokoro wrote:

All distances of the points on the sphere to be Integer

n=2, r=0.5, l=1 (0.5,0,0),(-0,5,0,0)

n=3,

The problem of the precise number that lies on the surface is of a number theoretic nature. It has to do with the number of ways we can express an integer as the sum of n squares. A lot of modern and classical work in number theory relates to this question.

The other problem, where we count all the points inside instead of just those on the boundary, is of a different flavour. If ar denotes the number of lattice points on the surface of the 3-d sphere with radius r centered at the origin, then each individual ar fluctuates quite erratically. If we study the sum a1+a2+⋯+ar instead, then we get smoother behavior and analytic methods can be applied.

19-Tanzanite
(To:CM_9836556)

Solved: All distances of the points to be Integer - PTC Community

Above is the one of 2D. I am solving the n points answer for both 2D and 3D.

If you know any study corresponds these puzzles, please let me know it.

Tokoro.

19-Tanzanite
(To:ttokoro)

This is not the answer of n=8.

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