First, we have the obvious solution of x = y = z = 0,
because (0 + 0 + 0)² = 0² = 0 = 0 + 0 + 0 = 0³ + 0³ + 0³;
second we have the obvious solution of x = -1, y = 1, z = 0,
because (-1 + 1 + 0)² = 0² = 0 = -1 + 1 + 0 = (-1)³ + 1³ + 0³
and permutations thereof by symmetry;
third we have the obvious solution of x = -1, y = 1, z = 1,
because (-1 + 1 + 1)² = 1² = 1 = -1 + 1 + 1 = (-1)³ + 1³ + 1³,
and permutations thereof by symmetry;
fourth we have the obvious solution of x = 1, y = 2, z = 3,
because (1 + 2 + 3)² = 6² = 36 = 1 + 8 + 27 = 1³ + 2³ + 3³,
and permutations thereof by symmetry;
fifth we have the obvious solution of x = 1, y = 2, z = 0,
because (1 + 2 + 0)² = 3² = 9 = 1 + 8 + 0 = 1³ + 2³ + 0³,
and permutations thereof by symmetry;
sixth we have the obvious solution of x = 1, y = 0, z = 0,
because (1 + 0 + 0)² = 1² = 1 = 1 + 0 + 0 = 1³ + 0³ + 0³,
and permutations thereof by symmetry; AND
seventh we have more generally (x+y)² + 2(x+y)z + z² = x³ + y³ + z³,
so x³ + y³ + z³ - (x+y)² - 2(x+y)z - z² = 0,
so z³ - z² - 2(x+y)z + (x³ + y³)* - (x+y)² = 0,
so z³ - z² - 2(x+y)z + (x+y)(x²-xy+y²)* - (x+y)² = 0,
so z³ - z² - 2(x+y)z + (x+y)(x²-xy+y²-(x+y)) = 0,
so z³ - z² - 2(x+y)z + (x+y)(x²-xy+y²-x-y) = 0,
using the cubic formula with the following we have:
a = 1, b = -1, c = -2(x+y), and d = (x+y)(x²-xy+y²-x-y)
z = ³√{q + ²√[q² + (r-p²)³]} + ³√{q² - ²√[q + (r-p²)³]} + p
p = -b/(3a), q = p³ + (bc-3ad)/(6a²), and r = c/(3a)
p = -(-1)/(3*1) = 1/3, q = (1/3)³ + (-1*-2(x+y)-3*1*d)/(6*1²), so
q = 1/27 + (2x+2y-3d)/6 = 2/54 + 9(2x+2y-3d)/54 = (2+18x+18y-27d)/54,
and r = -2(x+y)/(3*1) = (-2x-2y)/3, so we have that:
z = ³√{(2+18x+18y-27d)/54 + ²√[(2+18x+18y-27d)²/54² + ((-2x-2y)/3-(1/3)²)³]} + ³√{(2+18x+18y-27d)²/54² - ²√[(2+18x+18y-27d)/54 + ((-2x-2y)/3-(1/3)²)³]} + (1/3),
so z = ³√{(2+18x+18y-27d)/54 + ²√[(2+18x+18y-27d)²/54² + (6x+6y+1)³/9³]} + ³√{(2+18x+18y-27d)²/54² - ²√[54*(2+18x+18y-27d)/54² - (2x+2y+1)³/9³]} + (1/3),
so z = ³√{(2+18x+18y-27d)/54 + ²√[(2+18x+18y-27d)²/54² + 4(6x+6y+1)³/54²]} + ³√{(2+18x+18y-27d)²/54² - ²√[54(2+18x+18y-27d)/54² - 4(2x+2y+1)³/54²]} + (1/3),
so z = ³√{(2+18x+18y-27d)+²√[(2+18x+18y-27d)²+4(6x+6y+1)³]}/(3*³√(2)) + ³√{(2+18x+18y-27d)²-54*²√[54(2+18x+18y-27d)-4(2x+2y+1)³]}/(9*³√(4)) + (1/3), where d = (x+y)(x²-xy+y²-x-y).
* x² - xy + y²
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(x + y) | x³ + 0 + y³
-(x³ + x²y)
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- x²y + y³
-(- x²y - xy²)
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y³ + xy²
-(xy² + y³)
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0
