Integer solutions of $x^n+y^n=z^n+w^n $ [$(n,2,2)$ Lander, Parkin, Selfridge]

Question

I am wondering if integers of the form

$x^n\:+y^n\:=z^n\:+w^n$

have a solution if none of $x,\:y,\:z,\:or\:w$ are equal.

Answer 1

There's always the taxicab identity: $$ 12^3+1^3 = 10^3+9^3 $$

It is not known whether examples exist for all exponents. For fourth powers there is $$ 133^4 + 134^4 = 158^4 + 59^4 $$ but even for fifth powers no example is known (nor a proof that there are no examples). See the Lander, Parkin, Selfridge conjecture. This is the special case $(n,2,2)$.