TITLE: Infinitely many integer solutions to $X^4+Y^4-18Z^4= -16$
QUESTION [13 upvotes]: We found infinitely many integer solutions to 
$$X^4+Y^4-18Z^4= -16 \qquad (1)$$.
The interesting part in this diophantine equation is the sum of
the reciprocals of the degrees is $3/4 < 1$, which is related to
Vojta's more general abc conjecture.

Q1 Is this result trivial or known?

Consider the diophantine equation
$$ aX^n+bY^m+cZ^l=d \qquad (2)$$
where $1/n+1/m+1/l < 1$. Solution of (2) is trivial if almost always
$d \in \{aX_0^n,bY_0^m,cZ_0^l\}$ and the sum of the other two
monomials vanishes.

Q2 Is there (2) with infinitely many integer non-trivial solutions,
  except for generalization of (1) with $n=m=l=4$?

REPLY [26 votes]: I do not know if the result is trivial or known, but your result is fun
and follows from the identity
$$(x-1)^4+(x+1)^4+16=2(x^2+3)^2$$
so that you find infinitely many solutions by solving the Pell equation
$x^2-3z^2=-3$, so $x+z\sqrt{3}=\pm\sqrt3(2+\sqrt3)^n$.
You can of course replace $x-1$ and $x+1$ by $ax+b$ and $ax-b$ and find
infinitely many solutions to similar equations.