A famous open question, discussed several times on MathOverFlow, asks Which integers can be expressed as a sum of three cubes in infinitely many ways?. This is open even for $n=3$, that is, we do not know whether the equation $x^3+y^3+z^3=3$ has infinitely many integer solutions.
We also do not know whether the equation $x^4+y^4+1=z^2$ have a non-trivial solution, see https://math.stackexchange.com/questions/16887/
The difficulties of these questions are related to the fact that $1/3+1/3+1/3=1/4+1/4+1/2=1$, hence heuristics predict that there are infinitely many integer solutions but they are rare. For example, the smallest solution to the second equation may have about 48 digits (see link above).
The problem should be much easier with the powers $a,b,c$ such that $1/a+1/b+1/c>1$, in which case heuristics predict many solutions. This is the case when $(a,b,c)$ is (up to permutation) equal to $(2,2,k)$ for $k\geq 2$ or $(2,3,k)$ for $3\leq k \leq 5$.
The case $(2,2,k)$ reduces to equation of the form $x^2+y^2=z^k+n$, that is, we need to check if $P(z)=z^k+n$ is a sum of two squares infinitely often. There are many methods for this problem, see e.g. Representing $x^3-2$ as a sum of two squares
The case $(2,3,3)$ is the equation of the form $x^3+y^3=z^2+n$. After substitution $y=t-x$ for a new variable $t$, it reduces to $t^3 - 3 t^2 x + 3 t x^2=z^2+n$. For any fixed $t$ this equation is quadratic in $(x,z)$, can be reduced to Pell's equation, and, unless there are clear obstructions for solution existence, it should not be too difficult to find $t$ for which it has infinitely many integer solutions in $(x,z)$.
The next case is $(2,3,4)$, that is, equation of the form $$ x^4+y^3=z^2+n. $$ Are there known methods for proving that, for some given $n$, this equation has infinitely many integer solutions in $(x,y,z)$? As two concrete examples, is this true for $n=2$ and $n=-2$?
Some ideas:
If we could find polynomials $P,Q,R$ is one variable $u$ with integer coefficients such that $P(u)^4+Q(u)^3=R(u)^2+n$, we would be done. But are there methods for finding such polynomials better than blind computer search?
If we could find polynomials $P,Q,R$ is two variables $u,v$ such that $S=P^4+Q^3-R^2$ is quadratic of the form $S=au^2+buv+cv^2+$(linear terms) with $b^2-4ac>0$ and not a perfect square, we could then prove that $S(u,v)=n$ has infinitely many integer solutions in $(u,v)$.
If we could find polynomials $P,Q$ in one variable $u$ such that $P^4+Q^3-n$ factorises as $R(u)^2\cdot S(u)$ with quadratic $S(u)$, then we could check if $S(u)=t^2$ has infinitely many integer solutions in $(t,u)$. But, once again, how to find such $P,Q$ other than blind search? Also, are there heuristic that checks whether such $P,Q$ are expected to exist at all?
