Let us consider the equation $a^n+b^n=c^n$ for positive integers $a,b,c$ and $n\ge 2$. The $n=2$ case has a well-known and beautiful parametrization known as Pythagorean triples. Fermat's Last Theorem gives us that there are no solutions for $n\ge 3$.
It can be observed from the literature that Diophantine equations involving a sum of individual variables to some like power, i.e. $a_1^n+\cdots+a_k^n=b_1^n+\cdots+b_m^n$, are considered to be among the most "attractive" of problems.
Maybe with more controversy, it can be said that constructing (a family of) solutions is easier with a greater number of terms, thanks to the added flexibility. In this sense, Fermat's question of splitting a power into two same powers in the "tightest" question.
This brings us to the next tightest equations, which add one more term: For which $n\ge 2$ do $a^n + b^n + c^n = d^n$ or $a^n + b^n = c^n + d^n$ (call them $(1)$ and $(2)$ respectively) have solutions in positive integers, what are some solutions, and can we find all of them? We are looking at non-trivial solutions, so in $(2)$, $a,b$ are distinct from $c,d$.
Of course, families or full parametrizations represented by polynomials are the prettiest. I read somewhere that full ones exist for $n=2$ in each case, but a reference would be nice.
Now $n\ge 3$ is a murky area, as far as I can tell. One famous result is Elkies' counterexamples to Euler's quartic conjecture, so $(1)$ has infinitely many solutions for $n=4$. The Mathworld pages on 3rd powers and 4th powers say that families of solutions to $(1)$ are known for $n=3$, as well as for $(2)$ for $n=3,4$.
At this point, for larger $n$, there seem to be no known solutions to either equations; correct me otherwise please. So my question is:
What are references to results on larger $n$ for these two equations? Has it been proven that no integer solutions exist for some, probably large, $n$? Maybe even all sufficiently large $n$?
Best Answer
It is a folklore conjecture that for $n \geq 5$ that for any positive integer $k$ there are at most two pairs of positive integers such that $x^n + y^n = k$ for $n \geq 5$, and they are related by the obvious symmetry of swapping $x,y$. See for example this paper by Skinner and Wooley.
The state of the art regarding this matter is to investigate how dense off-diagonal solutions to the equation $x^n + y^n = u^n + v^n$ is. As far as I know this was done in general by Bennett, Dummigan, and Wooley in 1998. My self and Cam Stewart obtained the more general result concerning $F(x,y) = F(u,v)$ for general binary forms of degree $d \geq 3$ in this paper.