Find the set of integers $a$ such that the equation $2 a^{2}=7 k+2$ has an integer solution $k$.

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The problem can be restated as follows: find the integers a such that $2 a^{2} \equiv 2(\bmod 7)$. Such a relation depends only on the congruence of a modulo 7. We then check by hand that $2 \times 0^{2} \equiv 0(\bmod 7), 2 \times 1^{2} \equiv 2 \times 6^{2} \equiv 2(\bmod 7), 2 \times 2^{2} \equiv 2 \times 5^{2} \equiv 1$ $(\bmod 7)$ and $2 \times 3^{2} \equiv 2 \times 4^{2} \equiv 4(\bmod 7)$. It follows that $\frac{2 a^{2}-2}{7}$ is an integer if and only if a is of the form $7 \ell+1$ or $7 \ell-1$.