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?

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