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?
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}.
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Posted by Charlie
on 2012-06-10 20:29:42 |
computer solution for part 2 (spoiler)
