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.

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 A-Z
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 2012-06-02 13:08:08