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Ramanujan enforced. (Posted on 2010-08-02) Difficulty: 4 of 5
1,729 is the least integer which equals the sum of two positive cubes in two different ways (see "taxicab number" on the web):
1,729 = 12^3+1^3 and 1,729 = 10^3+9^3.

a) Find an integer which equals the sum of two positive cubes in THREE different ways.
There is more than one solution.

b) Erasing the word "positive" in the 1st statement allows a lesser positive integer(s??) number(s??) to be presented as a sum of two cubes in two different ways.
Find it (them??).

No Solution Yet Submitted by Ady TZIDON    
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Solution computer solution | Comment 2 of 5 |

Part A:


DEFDBL A-Z
OPEN "ramanuj.txt" FOR OUTPUT AS #2
FOR a = 1 TO 1000
 a3 = a * a * a
FOR b = a TO 1000
 b3 = b * b * b
  v = a3 + b3
  PRINT #2, USING "#### #### ###########"; a; b; v
NEXT
NEXT
CLOSE

Then, the ramanuj.txt file was sorted on the total value (v), and read by the following program:

OPEN "ramanuj.txt" FOR INPUT AS #1
CLS
DO
  pprevl$ = prevl$
  prevl$ = l$
  LINE INPUT #1, l$
  pprevn$ = prevn$
  prevn$ = n$
  n$ = MID$(l$, 12, 10)
  IF n$ = pprevn$ THEN
     IF flag = 0 THEN PRINT pprevl$: PRINT prevl$: ct = ct + 2
     flag = 1
     PRINT l$: ct = ct + 1
  ELSE
     IF flag THEN PRINT : ct = ct + 1
     IF ct > 40 THEN ct = 0: DO: LOOP UNTIL INKEY$ > ""
     flag = 0
  END IF
LOOP UNTIL EOF(1)
CLOSE

which found the following as the first few values that each had three ways of being produced by the sum of positive cubes (the cube roots of the cubes are shown, together with the sum of the cubes):

167  436    87539319
228  423    87539319
255  414    87539319
 11  493   119824488
 90  492   119824488
346  428   119824488
359  460   143604279
408  423   143604279
111  522   143604279
315  525   175959000
 70  560   175959000
198  552   175959000
300  670   327763000
339  661   327763000
510  580   327763000
510  828   700314552
334  872   700314552
456  846   700314552
295  920   804360375
 15  930   804360375
198  927   804360375
692  856   958595904
 22  986   958595904
180  984   958595904

Part B:

DEFDBL A-Z
OPEN "ramanub.txt" FOR OUTPUT AS #2
FOR a = -1000 TO 1000
 a3 = a * a * a
FOR b = ABS(a) TO 1000
 b3 = b * b * b
  v = a3 + b3
  PRINT #2, USING "##### ##### ###########"; a; b; v
NEXT
NEXT
CLOSE

Then, the ramanuB.txt file was sorted on the total value (v), and zero values eliminated, and read by the following program:

OPEN "ramanub.txt" FOR INPUT AS #1
CLS
DO
  pprevl$ = prevl$
  prevl$ = l$
  LINE INPUT #1, l$
  pprevn$ = prevn$
  prevn$ = n$
  n$ = MID$(l$, 14, 10)
  IF n$ = prevn$ THEN
     IF flag = 0 THEN PRINT prevl$: ct = ct + 1
     flag = 1
     PRINT l$: ct = ct + 1
  ELSE
     IF flag THEN PRINT : ct = ct + 1
     IF ct > 40 THEN ct = 0: DO: LOOP UNTIL INKEY$ > ""
     flag = 0
  END IF
LOOP UNTIL EOF(1)
CLOSE

Results include not only the first few sums of two cubes in two ways but also the first achievable in three ways: 728. The results are shown to the first one that's beyond Ramanujan's 1729:

  3     4          91
 -5     6          91
 -4     6         152
  3     5         152
 -3     6         189
  4     5         189
  1     6         217
 -8     9         217
 -6     9         513
  1     8         513
 -2     9         721
-15    16         721
-10    12         728
  6     8         728
 -1     9         728
 -9    12         999
 -1    10         999
-18    19        1027
  3    10        1027
 -8    12        1216
  6    10        1216
 -6    12        1512
  8    10        1512
  9    10        1729
  1    12        1729
-16    18        1736
  2    12        1736
  9    12        2457
-15    18        2457

  Posted by Charlie on 2010-08-02 16:03:48
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