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Cubic AND Quartic Challenge (Posted on 2004-02-06) Difficulty: 4 of 5
What is the smallest positive integer that is the sum of two different pairs of (non-zero, positive) cubes?
_____________________________

What is the smallest positive integer that is the sum of two different pairs of integers raised to the 4th power? and how did you find it?

In other words what is the smallest x such that:
x = a^4 + b^4 = c^4 + d^4
(where x, a, b, c, and d are all different, non-zero, positive integers)?
_____________________________

Are you able to determine the answer without looking it up on the internet?

See The Solution Submitted by SilverKnight    
Rating: 2.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution | Comment 2 of 15 |
This is a problem which does call for a brute-force program or two:

For the cube problem, the following program produces a file, which, when sorted, can be read by a second program to check for and report duplicates:


5 open "cubt.txt" for output as #2
10 for Sum=2 to 2000
20 Lim=int(Sum/2)
30 for A=1 to Lim
40 B=Sum-A
50 Tot=A*A*A+B*B*B
60 print #2,using(15,0),Tot;:print #2,A;B
70 next
80 next
90 close

The sort is done by
sort cubt.txt > cubts.txt
The second program is:

DEFDBL A-Z
CLS
OPEN "cubts.txt" FOR INPUT AS #1
DO
LINE INPUT #1, l$
n = VAL(LEFT$(l$, 15))
IF n = nPrev THEN
PRINT lPrev$
PRINT l$
PRINT
ct = ct + 1
IF ct / 12 = INT(ct / 12) THEN
END
END IF
END IF
nPrev = n
lPrev$ = l$
LOOP UNTIL EOF(1)

----
The first duplicate is for a^3+b^3 = 1729. The first few are:

1729 1 12
1729 9 10

4104 2 16
4104 9 15

13832 18 20
13832 2 24

20683 10 27
20683 19 24

32832 18 30
32832 4 32

39312 15 33
39312 2 34

40033 16 33
40033 9 34

46683 27 30
46683 3 36

64232 17 39
64232 26 36

65728 12 40
65728 31 33

110656 36 40
110656 4 48

110808 27 45
110808 6 48

----
The program for fourth powers is similar and finds the first such match at 635,318,657, as shown in the following output for the first few.

635318657 59 158
635318657 133 134

3262811042 7 239
3262811042 157 227

8657437697 193 292
8657437697 256 257

10165098512 118 316
10165098512 266 268

51460811217 177 474
51460811217 399 402

52204976672 14 478
52204976672 314 454

68899596497 271 502
68899596497 298 497

86409838577 103 542
86409838577 359 514

138519003152 386 584
138519003152 512 514

160961094577 222 631
160961094577 503 558

162641576192 236 632
162641576192 532 536

264287694402 21 717
264287694402 471 681

-------

The cube problem is famous for Ramanujan's notice of it in a taxicab with mathematician G.H.Hardy.
  Posted by Charlie on 2004-02-06 09:30:25
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