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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 $n$?

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The non-existence of solutions for $n$ large is an open problem. It's known that the corresponding surfaces contain only finitely many rational or elliptic curves for $n$ large ($\ge 16$?), so if you believe the Bombieri-Lang conjecture... – Felipe Voloch Jun 19 '13 at 18:48
Actually under Bombieri-Lang the number is already finite for $n \geq 5$, because that's when the surface is of general type. It is known unconditionally (via the Wronskian trick) that there are no nontrivial rational curves for $n \geq 8$, nor any elliptic curves for $n \geq 9$. There's a nontrivial rational curve for $n=5$ (residual conic on the intersection of $a^5+b^5+c^5+d^5=0$ with $a+b+c+d=0$), but unfortunately it has no rational or even real points. For $n=6$ there's an elliptic curve that yields infinitely many rational points on some twists such as $a^6+2b^6+125c^6=2d^6$. – Noam D. Elkies Jun 19 '13 at 19:00
For the side questions: for references for classical results you mention, the Survey of equal sums of like powers by Lander, Parkin, Selfridge could be of interest to you, see (it is free) Also for lists of solutions, in cases where they exist, for these types of equations (more variables) is a nice starting point linking to other resources such as the EulerNet a distributed computing effort. – user9072 Jun 19 '13 at 19:35
NB When I commented "the number is already finite" I meant the number of rational or elliptic curves other than the obvious lines such as the line $a=c,b=d$ on $a^n+b^n = c^n+d^n$. If that number is zero (as is known to be the case for $n \geq 9$), then the same conjecture implies that the number of nontrivial rational solutions is finite too. The expectation is that there are no nontrivial rational solutions for either equation once $n \geq 5$. – Noam D. Elkies Jun 19 '13 at 20:03
You can have a quick overview of known solvable cases in this related MO answer. – Tito Piezas III Jan 3 '15 at 6:21

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