All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Two possible sums (Posted on 2013-11-08)
Each of the numbers 3, 10, 17, 24, 29, 36, ... can be written as a sum of three cubes of positive integers - not necessarily all distinct.

What is the smallest member in this series that allows more than one way of such representation?

To avoid misunderstanding - distinct order in the summation of the cubes will not count as a different set - e.g. (8,1,1) , (1,8,1) and (1,1,8) represent the same triplet.

 No Solution Yet Submitted by Ady TZIDON No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 computer solution Comment 1 of 1

DEFDBL A-Z

OPEN "2posssum.txt" FOR OUTPUT AS #2
FOR a = 1 TO 150
PRINT a;
ac = a * a * a
FOR b = a TO 150
bc = b * b * b
FOR c = b TO 150
cc = c * c * c
sum = ac + bc + cc
PRINT #2, USING "######### ### ### ### ######### ######### #########"; sum; a; b; c; ac; bc; cc
NEXT
NEXT
NEXT

CLOSE

produces a file, which is then sorted and then read by:

DEFDBL A-Z
CLS
OPEN "2posssum.txt" FOR INPUT AS #1
DO
LINE INPUT #1, l\$
IF LEFT\$(l\$, 10) = LEFT\$(p\$, 10) THEN
PRINT p\$: PRINT l\$: PRINT
ct = ct + 1
IF ct > 10 THEN END
END IF
p\$ = l\$
LOOP UNTIL EOF(1)

which produces the first few pairs of the type requested:

`        bases of sum      cubes                 cubes  251   1   5   5         1       125       125 251   2   3   6         8        27       216`
`1009   4   6   9        64       216       7291009   1   2  10         1         8      1000`
`1366   2   3  11         8        27      13311366   5   8   9       125       512       729`
`1457   6   8   9       216       512       7291457   1   5  11         1       125      1331`
`1459   4   4  11        64        64      13311459   1   9   9         1       729       729`
`1520   2   8  10         8       512      10001520   4   5  11        64       125      1331`
`1730   1   9  10         1       729      10001730   1   1  12         1         1      1728`
`1737   1   2  12         1         8      17281737   2   9  10         8       729      1000`
`1756   1   3  12         1        27      17281756   3   9  10        27       729      1000`
`1763   2   3  12         8        27      17281763   6   6  11       216       216      1331`
`1793   1   4  12         1        64      17281793   4   9  10        64       729      1000`

so 251 is the first member of the series that allows more than one representation as the sum of positive cubes.

If the cubes needed to be distinct then the smallest would be 1009.

 Posted by Charlie on 2013-11-08 17:25:13

 Search: Search body:
Forums (0)