how to solve this equation using certain concepts

Question

Solve in $\mathbb{Z}$ the following equation:

$x^6$ + $3x^3$ + $1$ = $y^4$,

using, if it's possible, prime numbers & decomposition in prime factors concepts...

Thanks for your time!

Answer 1

Nice. Let $n = x^3$, then: $n^2 + 3n + 1 - y^4 = 0$.

$\triangle = 3^2 - 4\cdot 1\cdot (1-y^4) = k^2 \to 9 - 4 + 4y^4 = k^2 \to 5 = (k - 2y^2)(k + 2y^2)$. From this you can continue...