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 $n=m=l=4$?

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 $n=m=l=4$?