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Four special natural numbers (Posted on 2008-12-11) Difficulty: 2 of 5
Find all sets of four natural numbers such that the square of each of them, when added to the sum of the other three, again yields a perfect square.

See The Solution Submitted by pcbouhid    
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Some Thoughts computer exploration--is there a pattern to this? | Comment 6 of 8 |
 1  1  1  1
 1  5  5  5
 1  8  8  8
 6  6  11  11
 1  16  16  16
 1  21  21  21
 1  33  33  33
 1  40  40  40
 1  56  56  56
 1  65  65  65
 1  85  85  85
 1  96  96  96
 40  57  96  96
 1  120  120  120
 1  133  133  133
 1  161  161  161
 1  176  176  176

DEFDBL A-Z
CLS
FOR t = 4 TO 612
 FOR a = 1 TO t / 4
 FOR b = a TO (t - a) / 3
 FOR c = b TO (t - a - b) / 2
 d = t - a - b - c
  good = 1
  sq = a * a + b + c + d
  sr = INT(SQR(sq) + .5)
  IF sr * sr = sq THEN
    sq = a + b * b + c + d
    sr = INT(SQR(sq) + .5)
    IF sr * sr = sq THEN
      sq = a + b + c * c + d
      sr = INT(SQR(sq) + .5)
      IF sr * sr = sq THEN
        sq = a + b + c + d * d
        sr = INT(SQR(sq) + .5)
        IF sr * sr = sq THEN
         PRINT a; b; c; d
        END IF
      END IF
    END IF
  END IF

 NEXT
 NEXT
 NEXT
NEXT



  Posted by Charlie on 2008-12-12 00:17:29
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