In order that all the squares be positive, x must be the largest and z the smallest. Y can't be negative and therefore X can't be negative.
The following program tests those numbers that meet these criteria for X up to 2000, and finds no solutions.
Either one of my assumptions is wrong or the solutions involve very large numbers.
DEFDBL AZ
DECLARE FUNCTION isSq (x)
FOR x = 0 TO 2000
FOR y = 0 TO x  1
FOR z = 1  y 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(z + y) THEN
IF isSq(y  z) THEN
PRINT x, y, z
END IF
END IF
END IF
END IF
END IF
END IF
NEXT
NEXT
NEXT
FUNCTION isSq (x)
sr = INT(SQR(x) + .5)
IF sr * sr = x THEN isSq = 1: ELSE isSq = 0
END FUNCTION

Posted by Charlie
on 20120602 13:08:08 