Find all $n$ which $7(n^2 + n + 1)$ is perfect $4^{th}$ power. https://ift.tt/eA8V8J

Find all positive integer $n$ , which $7(n^2 + n + 1)$ is perfect $4^{th}$ power.

What I tried

Let $7(n^2 + n + 1) = a^4$ $\to$ $ 7 | a$ and $a$ is odd.

We then get $(n^2 + n + 1) = 343k^4$ ; $k \in \mathbb Z$

Hence, $ 343 | n^3 - 1$. I’m stuck here

Please help! Thanks in advance.

Ps : This problem is from my teacher , in the topic of polynomial and it’s application.

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