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The smallest sum II (Posted on 2012-06-10) Difficulty: 3 of 5

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    
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Solution 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;           z
x+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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