I am having trouble proving this. I know that I need to look at the quadratic residues mod 8 and that they are 1,4 or 0. However how do I prove that these are the equivalence classes of Z/3Z.

Then I can added the equivalence classes and show that none of them divide 8. How do I set this proof up?


We may be overthinking this. Residue $7$ modulo $8$ cannot be rendered as a sum of three residues each belonging to $\{0,1,4\}$. Proof: The sum can be odd only if there are an odd number of $1$'s; one $1$ gives a sum one greater than a multiple of $4$ and three $1$'s give $1+1+1\equiv 3$ not $7$. Done.

  • $\begingroup$ I don't understand where the residue 7 mod 8 comes from. Can someone explain? $\endgroup$ – mathamasacre Sep 29 '16 at 15:53
  • $\begingroup$ Because we need only one counterexample, and residue 7 is it. $\endgroup$ – Oscar Lanzi Sep 29 '16 at 16:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.