$\forall p \ge 11 \quad \exists n \in \{1,2,\dots,9\}$ such that $n$ and $n+1$ are quadratic residues mod p

Question

I must solve the following problem:

$\forall p \ge 11 \quad \exists n \in \{1,2,\dots,9\}$ such that $n$ and $n+1$ are quadratic residues mod $p$.

I read some questions here about this topic but I found them a bit different so I couldn't find my answer.

My attempt is the following:

• we know that a prime $p$ is $p \equiv 1 \pmod 4 $ or $p \equiv -1 \pmod 4 $, so I divided the two cases

• CASE 1: $p \equiv 1 \pmod 4 \longrightarrow (2|p)=1 $ and $(1|p)=1 \forall p$

• CASE 2: $p \equiv -1 \pmod 4$. It is sufficient to control the primes $q \in \{1,2,\dots,9\}$ because $(a|p)=1$ and $(b|p)=1 \longrightarrow (ab|p)=1$. So I must check $q=\{2,3,5,7\}$.

From here I am not able to continue. Have you any suggestions?

Answer 1

Hint: one of $2,5,10$ must be a quadratic residue.