Find $x, y$, and $z$ such that $x^3+y^3+z^3=k$, for each $k$ from $1$ to $100.$

Question

It’s called a Diophantine Equation, and it’s sometimes known as the “summing of three cubes”.

A Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is such that all the unknowns take integer values).

It seems easy for $x^3+y^3+z^3=8$. $x=1$, $y=-1$ and $z=2.$

But what for higher values of $k$?

Answer 1

Using positive integers only for $x,y,z$ there are no solutions for $k<3$ but here are $3\le k \le 100$. The use of non-positive integers follows.

 (x,y,z,k)
 (1,1,1,3)
 (1,1,2,10)
 (1,2,2,17)
 (2,2,2,24)
 (1,1,3,29)
 (1,2,3,36)
 (2,2,3,43)
 (1,3,3,55)
 (2,3,3,62)
 (1,1,4,66)
 (1,2,4,73)
 (2,2,4,80)
 (3,3,3,81)
 (1,3,4,92)
 (2,3,4,99)

It appears that at least one of x,y,z must be positive if k is positive.

 (x,y,z,k)
 (-100,1,100,1)
 (-99,1,99,1)
 (-98,1,98,1)
 (-97,1,97,1)
 (-96,1,96,1)
 (-95,1,95,1)
 (-94,1,94,1)
 (-93,1,93,1)
 (-92,1,92,1)
 (-91,1,91,1)
 (-90,1,90,1)
 (-89,1,89,1)
 (-88,1,88,1)
 (-87,1,87,1)
 (-86,1,86,1)
 (-85,1,85,1)
 (-84,1,84,1)
 (-83,1,83,1)
 (-82,1,82,1)
 (-81,1,81,1)
 (-80,1,80,1)
 (-79,1,79,1)
 (-78,1,78,1)
 (-77,1,77,1)
 (-76,1,76,1)
 (-75,1,75,1)
 (-74,1,74,1)
 (-73,1,73,1)
 (-72,1,72,1)
 (-71,1,71,1)
 (-70,1,70,1)
 (-69,1,69,1)
 (-68,1,68,1)
 (-67,1,67,1)
 (-66,1,66,1)
 (-65,1,65,1)
 (-64,1,64,1)
 (-63,1,63,1)
 (-62,1,62,1)
 (-61,1,61,1)
 (-60,1,60,1)
 (-59,1,59,1)
 (-58,1,58,1)
 (-57,1,57,1)
 (-56,1,56,1)
 (-55,1,55,1)
 (-54,1,54,1)
 (-53,1,53,1)
 (-52,1,52,1)
 (-51,1,51,1)
 (-50,1,50,1)
 (-49,1,49,1)
 (-48,1,48,1)
 (-47,1,47,1)
 (-46,1,46,1)
 (-45,1,45,1)
 (-44,1,44,1)
 (-43,1,43,1)
 (-42,1,42,1)
 (-41,1,41,1)
 (-40,1,40,1)
 (-39,1,39,1)
 (-38,1,38,1)
 (-37,1,37,1)
 (-36,1,36,1)
 (-35,1,35,1)
 (-34,1,34,1)
 (-33,1,33,1)
 (-32,1,32,1)
 (-31,1,31,1)
 (-30,1,30,1)
 (-29,1,29,1)
 (-28,1,28,1)
 (-27,1,27,1)
 (-26,1,26,1)
 (-25,1,25,1)
 (-24,1,24,1)
 (-23,1,23,1)
 (-22,1,22,1)
 (-21,1,21,1)
 (-20,1,20,1)
 (-19,1,19,1)
 (-18,1,18,1)
 (-17,1,17,1)
 (-16,1,16,1)
 (-15,1,15,1)
 (-14,1,14,1)
 (-13,1,13,1)
 (-12,1,12,1)
 (-12,9,10,1)
 (-11,1,11,1)
 (-10,1,10,1)
 (-9,1,9,1)
 (-8,-6,9,1)
 (-8,1,8,1)
 (-7,1,7,1)
 (-6,1,6,1)
 (-5,1,5,1)
 (-4,1,4,1)
 (-3,1,3,1)
 (-2,1,2,1)
 (-1,1,1,1)
 (0,0,1,1)
 (-47,-24,49,2)
 (-6,-5,7,2)
 (0,1,1,2)
 (-5,4,4,3)
 (1,1,1,3)
 (-58,-43,65,6)
 (-1,-1,2,6)
 (-1,0,2,7)
 (-100,2,100,8)
 (-99,2,99,8)
 (-98,2,98,8)
 (-97,2,97,8)
 (-96,2,96,8)
 (-95,2,95,8)
 (-94,2,94,8)
 (-93,2,93,8)
 (-92,2,92,8)
 (-91,2,91,8)
 (-90,2,90,8)
 (-89,2,89,8)
 (-89,41,86,8)
 (-88,2,88,8)
 (-87,2,87,8)
 (-86,2,86,8)
 (-85,2,85,8)
 (-84,2,84,8)
 (-83,2,83,8)
 (-82,2,82,8)
 (-81,2,81,8)
 (-80,2,80,8)
 (-79,2,79,8)
 (-78,2,78,8)
 (-77,2,77,8)
 (-76,2,76,8)
 (-75,2,75,8)
 (-74,2,74,8)
 (-73,2,73,8)
 (-72,2,72,8)
 (-71,2,71,8)
 (-70,2,70,8)
 (-69,2,69,8)
 (-68,2,68,8)
 (-67,2,67,8)
 (-66,2,66,8)
 (-65,2,65,8)
 (-64,2,64,8)
 (-63,2,63,8)
 (-62,2,62,8)
 (-61,2,61,8)
 (-60,2,60,8)
 (-59,2,59,8)
 (-58,2,58,8)
 (-57,2,57,8)
 (-56,2,56,8)
 (-55,2,55,8)
 (-54,2,54,8)
 (-53,2,53,8)
 (-52,2,52,8)
 (-51,2,51,8)
 (-50,2,50,8)
 (-49,2,49,8)
 (-48,2,48,8)
 (-47,2,47,8)
 (-46,2,46,8)
 (-45,2,45,8)
 (-44,2,44,8)
 (-43,2,43,8)
 (-42,2,42,8)
 (-41,2,41,8)
 (-40,-17,41,8)
 (-40,2,40,8)
 (-39,2,39,8)
 (-38,2,38,8)
 (-37,2,37,8)
 (-36,2,36,8)
 (-35,2,35,8)
 (-34,2,34,8)
 (-34,15,33,8)
 (-33,2,33,8)
 (-32,2,32,8)
 (-31,2,31,8)
 (-30,2,30,8)
 (-29,2,29,8)
 (-28,2,28,8)
 (-27,2,27,8)
 (-26,2,26,8)
 (-25,2,25,8)
 (-24,2,24,8)
 (-24,18,20,8)
 (-23,2,23,8)
 (-22,2,22,8)
 (-21,2,21,8)
 (-20,2,20,8)
 (-19,2,19,8)
 (-18,2,18,8)
 (-17,2,17,8)
 (-16,-12,18,8)
 (-16,2,16,8)
 (-16,9,15,8)
 (-15,2,15,8)
 (-14,2,14,8)
 (-13,2,13,8)
 (-12,2,12,8)
 (-11,2,11,8)
 (-10,2,10,8)
 (-9,2,9,8)
 (-8,2,8,8)
 (-7,2,7,8)
 (-6,2,6,8)
 (-5,2,5,8)
 (-4,2,4,8)
 (-3,2,3,8)
 (-2,2,2,8)
 (-1,1,2,8)
 (0,0,2,8)
 (0,1,2,9)
 (-3,-3,4,10)
 (1,1,2,10)
 (-2,-2,3,11)
 (-11,7,10,12)
 (-46,23,44,15)
 (-1,2,2,15)
 (-94,-48,98,16)
 (-12,-10,14,16)
 (0,2,2,16)
 (-52,25,50,17)
 (1,2,2,17)
 (-2,-1,3,18)
 (-95,47,91,19)
 (-77,26,76,19)
 (-16,-14,19,19)
 (-2,0,3,19)
 (-56,21,55,20)
 (-2,1,3,20)
 (-86,28,85,21)
 (-14,-11,16,21)
 (-10,8,8,24)
 (2,2,2,24)
 (-1,-1,3,25)
 (-1,0,3,26)
 (-100,3,100,27)
 (-99,3,99,27)
 (-98,3,98,27)
 (-97,3,97,27)
 (-96,3,96,27)
 (-95,3,95,27)
 (-94,3,94,27)
 (-93,3,93,27)
 (-92,3,92,27)
 (-91,3,91,27)
 (-90,3,90,27)
 (-89,3,89,27)
 (-88,3,88,27)
 (-87,3,87,27)
 (-86,3,86,27)
 (-85,3,85,27)
 (-84,3,84,27)
 (-83,3,83,27)
 (-82,3,82,27)
 (-81,3,81,27)
 (-80,3,80,27)
 (-79,3,79,27)
 (-78,3,78,27)
 (-77,3,77,27)
 (-76,3,76,27)
 (-75,3,75,27)
 (-74,3,74,27)
 (-73,3,73,27)
 (-72,3,72,27)
 (-71,3,71,27)
 (-70,3,70,27)
 (-69,3,69,27)
 (-68,3,68,27)
 (-67,3,67,27)
 (-66,3,66,27)
 (-65,3,65,27)
 (-64,3,64,27)
 (-63,3,63,27)
 (-62,3,62,27)
 (-61,3,61,27)
 (-60,3,60,27)
 (-60,22,59,27)
 (-59,3,59,27)
 (-58,3,58,27)
 (-57,3,57,27)
 (-56,3,56,27)
 (-55,3,55,27)
 (-54,3,54,27)
 (-53,3,53,27)
 (-52,3,52,27)
 (-51,3,51,27)
 (-50,3,50,27)
 (-49,3,49,27)
 (-48,3,48,27)
 (-47,3,47,27)
 (-46,3,46,27)
 (-45,3,45,27)
 (-44,3,44,27)
 (-43,3,43,27)
 (-42,3,42,27)
 (-41,3,41,27)
 (-40,3,40,27)
 (-39,3,39,27)
 (-38,3,38,27)
 (-37,-36,46,27)
 (-37,3,37,27)
 (-36,3,36,27)
 (-36,27,30,27)
 (-35,3,35,27)
 (-34,3,34,27)
 (-33,3,33,27)
 (-32,3,32,27)
 (-31,3,31,27)
 (-30,3,30,27)
 (-29,3,29,27)
 (-28,3,28,27)
 (-27,3,27,27)
 (-26,3,26,27)
 (-25,3,25,27)
 (-24,-18,27,27)
 (-24,3,24,27)
 (-23,3,23,27)
 (-22,3,22,27)
 (-21,3,21,27)
 (-20,3,20,27)
 (-19,3,19,27)
 (-18,-10,19,27)
 (-18,3,18,27)
 (-17,3,17,27)
 (-16,3,16,27)
 (-15,3,15,27)
 (-14,3,14,27)
 (-13,3,13,27)
 (-12,3,12,27)
 (-11,3,11,27)
 (-10,3,10,27)
 (-9,3,9,27)
 (-8,3,8,27)
 (-7,3,7,27)
 (-6,3,6,27)
 (-5,-4,6,27)
 (-5,3,5,27)
 (-4,3,4,27)
 (-3,3,3,27)
 (-2,2,3,27)
 (-1,1,3,27)
 (0,0,3,27)
 (-59,31,56,28)
 (-17,13,14,28)
 (0,1,3,28)
 (-20,13,18,29)
 (-3,-2,4,29)
 (1,1,3,29)
 (-6,5,5,34)
 (-4,-3,5,34)
 (-1,2,3,34)
 (-13,-8,14,35)
 (0,2,3,35)
 (-75,40,71,36)
 (-3,-1,4,36)
 (1,2,3,36)
 (-56,37,50,37)
 (-3,0,4,37)
 (-27,16,25,38)
 (-3,1,4,38)
 (-52,20,51,43)
 (-13,8,12,43)
 (-7,-7,9,43)
 (2,2,3,43)
 (-7,-5,8,44)
 (-3,2,4,45)
 (-29,19,26,46)
 (-2,3,3,46)
 (-50,-50,63,47)
 (-30,-14,31,47)
 (-8,6,7,47)
 (-26,-23,31,48)
 (-2,-2,4,48)
 (-4,-2,5,53)
 (-1,3,3,53)
 (-18,-15,21,54)
 (-11,-7,12,54)
 (0,3,3,54)
 (-23,-23,29,55)
 (-9,-6,10,55)
 (-2,-1,4,55)
 (1,3,3,55)
 (-47,31,42,56)
 (-21,-11,22,56)
 (-2,0,4,56)
 (-38,25,34,57)
 (-2,1,4,57)
 (-4,-1,5,60)
 (-4,0,5,61)
 (-43,22,41,62)
 (-34,27,27,62)
 (-4,1,5,62)
 (-1,-1,4,62)
 (2,3,3,62)
 (-63,-37,67,63)
 (-58,-38,63,63)
 (-6,-4,7,63)
 (-1,0,4,63)
 (-100,4,100,64)
 (-99,4,99,64)
 (-98,4,98,64)
 (-97,4,97,64)
 (-96,4,96,64)
 (-95,4,95,64)
 (-94,4,94,64)
 (-93,4,93,64)
 (-92,4,92,64)
 (-91,4,91,64)
 (-90,4,90,64)
 (-89,4,89,64)
 (-88,4,88,64)
 (-87,4,87,64)
 (-86,4,86,64)
 (-85,4,85,64)
 (-84,4,84,64)
 (-83,4,83,64)
 (-82,4,82,64)
 (-81,4,81,64)
 (-80,-34,82,64)
 (-80,4,80,64)
 (-79,4,79,64)
 (-78,4,78,64)
 (-77,4,77,64)
 (-76,4,76,64)
 (-75,4,75,64)
 (-74,4,74,64)
 (-73,4,73,64)
 (-72,4,72,64)
 (-71,4,71,64)
 (-70,4,70,64)
 (-69,4,69,64)
 (-68,4,68,64)
 (-68,30,66,64)
 (-67,4,67,64)
 (-66,4,66,64)
 (-65,4,65,64)
 (-64,4,64,64)
 (-63,4,63,64)
 (-62,4,62,64)
 (-61,4,61,64)
 (-60,4,60,64)
 (-59,4,59,64)
 (-58,4,58,64)
 (-57,4,57,64)
 (-56,4,56,64)
 (-55,4,55,64)
 (-54,4,54,64)
 (-53,4,53,64)
 (-52,4,52,64)
 (-51,4,51,64)
 (-50,4,50,64)
 (-49,4,49,64)
 (-48,4,48,64)
 (-48,36,40,64)
 (-47,4,47,64)
 (-46,4,46,64)
 (-45,4,45,64)
 (-44,4,44,64)
 (-43,4,43,64)
 (-42,4,42,64)
 (-41,4,41,64)
 (-40,4,40,64)
 (-39,4,39,64)
 (-38,4,38,64)
 (-37,4,37,64)
 (-36,4,36,64)
 (-35,4,35,64)
 (-34,4,34,64)
 (-33,4,33,64)
 (-32,-24,36,64)
 (-32,4,32,64)
 (-32,18,30,64)
 (-31,4,31,64)
 (-30,4,30,64)
 (-29,4,29,64)
 (-28,4,28,64)
 (-27,4,27,64)
 (-26,4,26,64)
 (-25,4,25,64)
 (-24,4,24,64)
 (-23,4,23,64)
 (-22,-17,25,64)
 (-22,4,22,64)
 (-21,4,21,64)
 (-20,4,20,64)
 (-19,4,19,64)
 (-18,4,18,64)
 (-17,4,17,64)
 (-16,4,16,64)
 (-15,4,15,64)
 (-14,4,14,64)
 (-13,4,13,64)
 (-12,4,12,64)
 (-11,4,11,64)
 (-10,4,10,64)
 (-9,4,9,64)
 (-8,4,8,64)
 (-7,4,7,64)
 (-6,4,6,64)
 (-5,-3,6,64)
 (-5,4,5,64)
 (-4,4,4,64)
 (-3,3,4,64)
 (-2,2,4,64)
 (-1,1,4,64)
 (0,0,4,64)
 (0,1,4,65)
 (1,1,4,66)
 (-22,-19,26,69)
 (-4,2,5,69)
 (-64,23,63,70)
 (-21,11,20,70)
 (-33,-22,36,71)
 (-24,12,23,71)
 (-20,-16,23,71)
 (-3,-3,5,71)
 (-1,2,4,71)
 (-27,-13,28,72)
 (-10,7,9,72)
 (0,2,4,72)
 (-47,29,43,73)
 (-24,-12,25,73)
 (1,2,4,73)
 (-55,26,53,78)
 (-66,-49,74,79)
 (-33,-19,35,79)
 (-6,-6,8,80)
 (2,2,4,80)
 (-18,10,17,81)
 (-15,12,12,81)
 (3,3,3,81)
 (-11,-11,14,82)
 (-36,-32,43,83)
 (-29,22,24,83)
 (-23,-15,25,83)
 (-5,-2,6,83)
 (-2,3,4,83)
 (-16,-9,17,88)
 (-4,-4,6,88)
 (-4,3,5,88)
 (-7,6,6,89)
 (-100,31,99,90)
 (-73,-32,75,90)
 (-51,-35,56,90)
 (-48,19,47,90)
 (-27,13,26,90)
 (-9,-8,11,90)
 (-5,-1,6,90)
 (-3,-2,5,90)
 (-1,3,4,90)
 (-5,0,6,91)
 (0,3,4,91)
 (-8,-5,9,92)
 (-5,1,6,92)
 (1,3,4,92)
 (-5,-5,7,93)
 (-22,14,20,96)
 (-22,17,18,97)
 (-3,-1,5,97)
 (-15,9,14,98)
 (-3,0,5,98)
 (-37,16,36,99)
 (-5,2,6,99)
 (-3,1,5,99)
 (2,3,4,99)
 (-6,-3,7,100)

Answer 2

This is the low-tech approach I took for this question. Note the optional output file in lines 140 and 220. The user may copy/paste or import the data into a spreadsheet for sorting as I did for the other answer.

 100 print "Enter limit";
 110 input l1
 120 print time$
 130 print
 140 rem   open "outfile.txt" for output as #1
 150 for n1 = -l1 to l1
 160     for n2 = n1 to l1
 170        for n3 = n2 to l1
 180            t0 = n1^3+n2^3+n3^3
 190            for t9 = 1 to l1
 200                if t0 = t9
 210                   print "    " n1 n2 n3 t9 "."
 220                   rem  print #1, n1 n2 n3 t9 "."
 230               endif
 240             next t9
 250         next n3
 260     next n2
 270         rem The following IF/ELSE skips zero
 280         rem     for the next iteration of n1
 290         if n1 = -1
 300            n1 = 1
 310            goto 160
 320         endif
 330 next n1
 340 print time$