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Lattice Vertices Verification (Posted on 2016-07-02) Difficulty: 3 of 5
The side lengths of a polygon with 1994 sides are given by:
A(n) = √(n2+4) for n= 1,2,…1994

Can all the vertices of this polygon lie on lattice points?

Give reasons for your answer.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts re: Solution | Comment 2 of 4 |
(In reply to Solution by Brian Smith)

I understand that the 3988-gon perimeter can be calculated in this way as follows with c^2={5,8,13,20,...}

1+2+2^2, 2^2+2^2, 2^2+3^2, 2^2+4^2 etc giving the result as claimed; (1993*2+1)+((1994*1995)/2+1) = 1993003.

But other representations are possible. For example, c^2=85=2^2+9^2 but also c^2=85=6^2+7^2, with a different perimeter. 1300 has 3 different representations {2,36}{12,34}{20,30}, and so on.

Also, why does the perimeter have to be 'even', when you make a convincing case that it has to be odd?!

Edited on July 3, 2016, 6:37 am
  Posted by broll on 2016-07-03 06:36:06

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