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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 dx.doi.org/10.1090/S0025-5718-1967-0222008-0 (it is free) Also for lists of solutions, in cases where they exist, for these types of equations (more variables) maa.org/editorial/mathgames/mathgames_11_13_06.html is a nice starting point linking to other resources such as the EulerNet euler.free.fr a distributed computing effort. –  quid 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 at 6:21

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