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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 $n=m=l=4$?

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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.

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  • $\begingroup$ Indeed, I use something very similar. $\endgroup$
    – joro
    Commented Apr 8, 2018 at 10:53
  • $\begingroup$ Yes. Did you find the identity by examining the numerical solutions? $\endgroup$
    – joro
    Commented Apr 8, 2018 at 12:55
  • $\begingroup$ Of course this is what I did first. $\endgroup$
    – Henri Cohen
    Commented Apr 8, 2018 at 17:33

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