is there an odd prime p such that (n/p)=1?

Question

For any positive integer n, is there exists an odd prime p such that $(\dfrac{n}{p}) = 1$?

i think that it will be true.. but i don't know what is right.

what i have done is that .. when n is even, power of 2 can be deprived form n and

we can make odd number b. so $(\dfrac{b}{p})$ can be legendre symbol.

i want to know the fact about that and the reason too

Answer 1

Yes. Pick your favorite square $m^2$, where $m$ is relatively prime to $n$ and has opposite parity to $n$; then any prime $p$ dividing $m^2-n$ is odd and has the property that $\big(\frac np\big) = \big(\frac {m^2}p\big) = \big(\frac mp\big)^2=1$.