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 The smallest sum II (Posted on 2012-06-10)

See The smallest sum.

1. Show that there are no positive integers {X,Y,Z} (with Z less than Y less than X) such that X+Y, X-Y, X+Z, X-Z, Y+Z, Y-Z are all squares; or provide a counter-example.

2. Assuming that no counter-example exists, what is the minimum such set {X,Y,Z} for which each of X+Y, X-Y, X+Z, X-Z, and either of Y+Z or Y-Z, are all squares?

 See The Solution Submitted by broll No Rating

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 computer solution for part 2 (spoiler) | Comment 1 of 3

DEFDBL A-Z
DECLARE FUNCTION isSq (x)
OPEN "small sum 3.txt" FOR OUTPUT AS #2
FOR x = 0 TO 1000
FOR y = 0 TO x - 1
FOR z = 1 TO y - 1
IF isSq(x + y) THEN
IF isSq(x - y) THEN
IF isSq(x + z) THEN
IF isSq(x - z) THEN
IF isSq(y + z) OR isSq(y - z) THEN

PRINT x, y, z
PRINT x + y; y + z; x + z; x - y; y - z; x - z
PRINT #2, x, y, z
PRINT #2, x + y; y + z; x + z; x - y; y - z; x - z

END IF
END IF
END IF
END IF
END IF
NEXT
NEXT
NEXT
CLOSE #2

FUNCTION isSq (x)
sr = INT(SQR(x) + .5)
IF sr * sr = x THEN isSq = 1: ELSE isSq = 0
END FUNCTION

finds

`pairs of rows are:x;              y;           zx+y; y+z; x+z; x-y; y-z; x-z`
` 125           100           44  225  144  169  25  56  81   500           400           176  900  576  676  100  224  324   533           308           92  841  400  625  225  216  441   650           506           250  1156  756  900  144  256  400   697           672           528  1369  1200  1225  25  144  169   725           644           500  1369  1144  1225  81  144  225   850           750           174  1600  924  1024  100  576  676   962           638           62  1600  700  1024  324  576  900 `

So the smallest {x,y,z} is {125,100,44}.

 Posted by Charlie on 2012-06-10 20:29:42
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