TITLE: Infinitely many integer solutions for the equations $x^3+y^3+z^3=1$ and $x^3+y^3+z^3=2$
QUESTION [5 upvotes]: How do you show that the equation $x^3+y^3+z^3=1$ has infinitely many solutions in integers? How about $x^3+y^3+z^3=2$?

REPLY [2 votes]: For
\begin{align}
x^3+y^3+z^3&=1\tag{1}
\end{align}
there are at least these one-parameter families of solutions
\begin{gather}
(9t^4)^3 + (-9t^4 \mp 3t)^3 + (\pm 9t^3 + 1)^3 = 1\tag{2}\\
(3888t^{10} - 135t^4)^3 + (-3888t^{10} - 1296t^7 - 81t^4 + 3t)^3 + (3888t^9 + 648t^6 - 9t^3 + 1)^3 = 1\\
\implies (27tu(144u^2-5))^3 + (-3t(27u(16u(3u+1)+1)-1))^3 + (9u(72u(6u+1)-1) + 1)^3 = 1, u=t^3\tag{3}\\
(-1679616a^{16}-559872a^{13}-27216a^{10}+3888a^7+63a^4-3a)^3 +\\
+ (1679616a^{16}-66096a^{10}+153a^4)^3 +\\
+ (1679616a^{15}+279936a^{12}-11664a^9-648a^6+9a^3+1)^3 = 1\\
\implies (3(-559872u^5-186624u^4-9072u^3+1296u^2+21u-1)a)^3 +\\
+ (9(186624u^4-7344u^2+17)au)^3 +\\
+ (1679616u^5+279936u^4-11664u^3-648u^2+9u+1)^3 = 1, u=a^3\tag{4}
\end{gather}
Beck attributes $(2)$ to Mahler (1936). For $(3)$ see mathpages. $(4)$ is the result of setting $b=1$ in an identity due to Kohmoto, cited by Wolfram, "Algebraic Identity".
Solutions to $(1)$ are a special case of "Fermat near-misses". Some are given in OEIS:
\begin{align}
A050787_i^3-A050789_i^3-A050788_i^3&=1\\
-A050791_i^3+A050793_i^3+A050792_i^3&=1
\end{align}
References:
Michael Beck et al. New integer representations as the sum of three cubes. Mathematics of Computation, 76, no.259 (Jul 2007), p.1683-1690. S 0025-5718(07)01947-3
Kurt Mahler. Note On Hypothesis K of Hardy and Littlewood. Journal of the London Mathematical Society, 11 (1936), p.136–138.