In order to obtain infinite integer non trivial solutions of the equation $X^4-DY^4=Z^4$ (all numbers natural) we do the following. We set $X=(r_1●p+p)$, $Y=p$, $Z=(r_1●p)$, $D=(4r_1^3+6r_1^2+4r_1+1)$, where $r_1$ is one rational solution of a particular cubic equation.
Let’s have the equation $(f●a+a)^4-(4f^3+6f^2+4f+1)●a^4=(f●a)^4$.
In the quantity $4f^3+6f^2+4f+1$ we set $f=k-1/2$ and we obtain the cubic $4k^3+k$. We equate this cubic with any integer $±m$ and we obtain $4k^3+k=±m$. Let’s say $r_1=b/p$ is one rational solution of this cubic equation. We set the value $b/p$ at $X, Y, Z, D$ and we obtain non trivial solutions of the above equation.
Let’s have $4k^3+k=15$ so $r_1=3/2$ and $5^4-34●2^4=3^4$. It is easily shown that the cubic $4k^3+k=±m$ has infinite rational solutions.
I tried to find a technique to obtain infinite solutions of the equation $X^5-DY^5=Z^5$ by applying the method of binomial separation but I could not find one. Does anyone know if such a technique exists?