$p \geq$ 7 show $\Big( \frac{n}{p} \Big)=\Big( \frac{n+1}{p} \Big) =1$ for atleast one n in the set $\{ 1,2,\ldots,,9 \}$

Question

Suppose that p is prime $\geq$ 7.Show that $\Big( \frac{n}{p} \Big)=\Big( \frac{n+1}{p} \Big) =1$ for atleast one n in the set $\{ 1,2,3,\ldots,9 \}$. I have read this understood For any prime p>5 proving the existence of consecutive quadratic residues and consecutive quadratic non residues How to do this how to show for at least one n ?

Answer 1

Suppose it were false and try to get a contradiction. For example, since $(\frac{1}{p}) = 1$, you would need $(\frac{2}{p}) = -1$. Since $(\frac{4}{p}) = 1$, what does that tell you about $(\frac{3}{p})$? Then keep going and remember the Legendre symbol is multiplicative.