All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 The smallest sum (Posted on 2012-06-02)
Find three distinct integers, X, Y and Z, such that X + Y, X + Z, Y + Z, X - Y, X - Z, and Y - Z are all squares of integers.
Apparently, there are many solutions.

Find the set [X, Y, Z] with the smallest X + Y + Z.

Comments: ( Back to comment list | You must be logged in to post comments.)
 aha -- computer solution | Comment 2 of 8 |
(In reply to thoughts by Charlie)

I neglected the possibility that Z might be exactly -Y, allowing for zero as a perfect square.

Each line shows X, Y and Z, with the line after showing the resulting squares, for X under 1000, showing the smallest set as [17,8,-8], with a total of 17:

17            8            -8
25  0  9  9  16  25
68            32           -32
100  0  36  36  64  100
82            18           -18
100  0  64  64  36  100
97            72           -72
169  0  25  25  144  169
153           72           -72
225  0  81  81  144  225
257           32           -32
289  0  225  225  64  289
272           128          -128
400  0  144  144  256  400
328           72           -72
400  0  256  256  144  400
337           288          -288
625  0  49  49  576  625
388           288          -288
676  0  100  100  576  676
425           200          -200
625  0  225  225  400  625
612           288          -288
900  0  324  324  576  900
626           50           -50
676  0  576  576  100  676
641           200          -200
841  0  441  441  400  841
706           450          -450
1156  0  256  256  900  1156
738           162          -162
900  0  576  576  324  900
833           392          -392
1225  0  441  441  784  1225
873           648          -648
1521  0  225  225  1296  1521
881           800          -800
1681  0  81  81  1600  1681

Edited on June 2, 2012, 1:58 pm
 Posted by Charlie on 2012-06-02 13:48:42

 Search: Search body:
Forums (0)