Let $p$ be an odd prime and $a$ be an integer that is not a perfect square. We can impose a congruence condition on $p$ that guarantees the equation $x^2=a \pmod p$ does not have a solution. For example, if $a=5$, then the condition $p \equiv 2 \pmod 5$ implies that 5 is a quadratic nonresidue modulo $p$. Similarly, one can impose a congruence condition on $p$ that guarantees a solution exists. Can this be generalized for other exponents. In other words:
Question 1: For what values of $n$ the following is true? Given any integer $a$ that is not the $n$th power of any integer, there exist integers $b,t$, such that for any prime $p$ with $p \equiv b \pmod t$ the equation $x^n \equiv a \pmod p$ does not have a solution.
Question 2: For what values of $n$ the following is true? Given any integer $a$, there exist integers $b,t$, such that for any prime $p$ with $p \equiv b \pmod t$ the equation $x^n \equiv a \pmod p$ has a solution.