Can I show that a positive power of $2$ plus a perfect square is not divisible by $7$?

Question

I've gotten another proof down to just having to prove that a positive power of $2$ plus a perfect square is not divisible by $7$:

$$7\nmid(2^k + n^2), \; k,n \in \mathbb{N}$$

How can I do this?

Answer 1

Edit:

Note that $$2^k \pmod{7} \in \{1,2,4\}$$

$$n^2 \pmod{7} \in \{0,1,2, 4 \}$$

We have listed the values that $2^k \pmod7$ can take, we just have to make sure that the values that $−2^k \pmod7$ can take is not in those values that $n^2 \pmod7 $ can take. For example, $1$ is a value that $2^k \pmod7$ can attain, $−1 \equiv 6 \pmod7$ is not a value that $n^2 \pmod7$ can take.

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Remark:

The original question was $7 \not\mid 2k+n^2$.

You can't since it is not true.

$$1^2+2(3)=7$$

A general strategy to get more counterexamples is choose any $n$, note the parity of $n$. find $l$ such that $7l > n^2$ where $l$ and $n$ share the same parity. $7l-n^2$ is a positive even number.