Complete residue set for $7$ where every residue is a square

Question

Exist a complete residue set for $7$ whose elements are all squares?

I tried to find them all by squaring a few positive integers but couldn’t find all the square residues module $7$. The numbers I couldn’t find were $3,5,6$

I think that the answer of the question is no but I haven’t been able to show it. If $n^2 \not \equiv 3 \pmod 7$ is true then it would be proved, but I don’t know how to prove this either.

Thanks in advance.

Answer 1

Since an integer $n$ is congruent $$n\equiv 0,\pm1, \pm2,\pm3 (\mod 7) $$ $(\mod 7)$, when squared you get $n^2$ congruent $$ n^2 \equiv 0,1,4,2 (\mod 7)$$

Thus it is not possible to have a whole residue system $(\mod 7)$made of squares.