Diophantine Equations of the form $ax^3+by^3+cz^3=dxyz$

Question

Let $a,b,c,d \in \mathbb{N}$ be constant positive integers such that $d^3 > abc$. Consider the Diophantine Equation of the form: $$ax^3+by^3+cz^3=dxyz \quad (x,y,z \in \mathbb{N})$$ Is there any known way to prove the non-existance of solutions for any $(a,b,c,d)$ if there are no modular restrictions $\big[$i.e. $\forall$ $n \in \mathbb{N}$ ; $\exists$ $(x,y,z)$ s.t. $ax^3+by^3+cz^3 \equiv dxyz \pmod{n}$$\big]$?

Answer 1

You can find some results about in Diophantine Equations of the great L.J.Mordell, Academic Press London and New York (1969). For example.

► $(7a+1)x^3+7b+2)y^3+(7c+4)z^3+(7d+1)xyz=0$ whit $(x,y,z)=1$ has only the trivial solution.

►Let $a,b,c$ be square-free integers, relatively prime in pairs, $abc\ne0$. Then the curve $$ax^3+by^3+cz^3+dxyz=0$$ when at most one of $a,b,c$ is $\pm1$, has either none or an infinity of rational points.

►There are other results in relation with the field $\mathbb Q(\sqrt{-3})$.

Answer 2

Above equation shown below:

$ax^3+by^3+cz^3=dxyz$ -----$(1)$

Equation $(1)$ has numerical solution given below:

$(x,y,z)=(23,17,7)$

$(a,b,c,d)=(1,1,30,10)$

Above is arrived from the identity given by

"J. Sylvester" & is shown below after some modification,

$x=(f^2g+g^2h+h^2f-3fgh)$

$y=(fg^2+gh^2+hf^2-3fgh)$

$z=(f^2+g^2+h^2-fg-fh-gh)$

$(a,b,c,d)=[(1),(1),(fgh),(f+g+h)]$

For, $(f,g,h)=(5,3,2)$ we get the numerical solution,

shown above for equation $(1)$.

Sylvester's Identity can be seen on "Tito Piezas"

website "Collection of Algebraic Identities" in

the third power section.