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The smallest sum (Posted on 2012-06-02) Difficulty: 3 of 5
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.

No Solution Yet Submitted by Ady TZIDON    
Rating: 4.6667 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution 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

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