TITLE: Quintic diophantine equation
QUESTION [8 upvotes]: How can I find non trivial primitive integer solutions, to the Diophantine equation $$a^4+b^4+c^4=d^5$$
Can anyone find me solutions to this equation? 
Or if possible a parametric equation that generates solutions?
I would appreciate any help
Ive also simplified it to finding coprime integer solutions greater then 1 to the equation,$$xyz(x^2+y^2+z^2)=1250w^5$$
I don't know if that helps at all.

REPLY [6 votes]: We can use the identity,
$$(2p-2q)^4+(2p+2q)^4+(4q)^4 = 2^5(p^2+3q^2)^2\tag1$$
One can then solve,
$$p^2+3q^2 = (a^2+3b^2)^k$$
for any $k$. For $k=5$, it is,
$$p =a^5 - 30 a^3 b^2 + 45 a b^4$$
$$q=b (5 a^4 - 30 a^2 b^2 + 9 b^4)$$
though $(1)$ has the common factor $2$.