Express each integer between 1 and 1000 as a sum of four cubes of integers (which can be positive, negative, or zero).

(In reply to Computer findings, leaving 45 cases for analytic methods by Charlie)

31 of the missing 45 are found by:

DEFDBL A-Z

DIM cube(50)

OPEN "sumcubes extra.txt" FOR OUTPUT AS #2
DATA 31,50,86,95,148,221,239,265,284,291,292,301,310,356,365,373,382,417,428
DATA 437,499,508,564,571,580,589,598,626,634,643,652,724,725,760,769,778
DATA 787,788,879,886,905,913,914,949,967
DIM needed(45)
FOR i = 1 TO 45
    READ needed(i)
NEXT

FOR n = 0 TO 50
    cube(n) = n * n * n: PRINT n, cube(n)
NEXT
FOR asub = 0 TO 50
    FOR bsub = asub TO 50
        FOR csub = bsub TO 50
            FOR dsub = csub TO 50
                FOR a = -cube(asub) TO cube(asub) STEP cube(asub)
                    FOR b = -cube(bsub) TO cube(bsub) STEP cube(bsub)
                        FOR c = -cube(csub) TO cube(csub) STEP cube(csub)
                            FOR d = -cube(dsub) TO cube(dsub) STEP cube(dsub)
                                tot = a + b + c + d
                                good = 0
                                FOR i = 1 TO 45
                                    IF needed(i) = tot THEN good = 1: needed(i) = 0: EXIT FOR
                                NEXT
                                IF good THEN
                                    IF tot > 0 AND tot <= 1000 THEN
                                        PRINT USING "#### = ###### + ####### + ####### + #######"; tot; a; b; c; d
                                        PRINT #2, USING "#### = ###### + ####### + ####### + #######"; tot; a; b; c; d
                                        ct = ct + 1
                                    END IF
                                END IF
                                IF dsub = 0 THEN EXIT FOR
                            NEXT
                            IF csub = 0 THEN EXIT FOR
                        NEXT
                        IF bsub = 0 THEN EXIT FOR
                    NEXT
                    IF asub = 0 THEN EXIT FOR
                NEXT
            NEXT
        NEXT
    NEXT
NEXT
PRINT ct

CLOSE

  50 =  24389 +   24389 +   68921 + -117649
  86 =   2744 +    2744 +   24389 +  -29791
  95 =     -1 +    2744 +    8000 +  -10648
 221 =     -1 +    8000 +   42875 +  -50653
 239 =     -1 +   -2197 +  -21952 +   24389
 291 =   1000 +    1728 +   21952 +  -24389
 292 =  -1331 +   -4913 +  -32768 +   39304
 301 =     -8 +  -17576 +  -32768 +   50653
 310 =      1 +  -17576 +  -32768 +   50653
 365 =    125 +   -2197 +  -21952 +   24389
 373 =      1 +    6859 +   97336 + -103823
 417 =     -8 +   -2744 +  -32768 +   35937
 428 =    -64 +   -6859 + -117649 +  125000
 437 =   -343 +   -2197 +  -29791 +   32768
 499 = -17576 +  -24389 +  -54872 +   97336
 508 =   1000 +    6859 +  117649 + -125000
 564 =     -1 +    4913 +    4913 +   -9261
 589 =     64 +    4096 +   39304 +  -42875
 634 =   6859 +    6859 +   29791 +  -42875
 643 =   -125 +   21952 +   64000 +  -85184
 652 =    343 +  -17576 +  -32768 +   50653
 760 =     -8 +   21952 +   64000 +  -85184
 769 =      1 +   21952 +   64000 +  -85184
 778 =     64 +   -2744 +  -12167 +   15625
 788 =      8 +   -2197 +  -29791 +   32768
 879 =    -64 +    2744 +   13824 +  -15625
 886 =     64 +   -4913 +   -4913 +   10648
 905 =    125 +   -2197 +  -29791 +   32768
 913 =  -4913 +   -8000 + -103823 +  117649
 914 =  -2197 +   -2197 +   -6859 +   12167
 967 =   4096 +   21952 +   29791 +  -54872

Posted by Charlie on 2013-10-23 12:55:41