# In the QS method of integer factoring, how does one know what number can be in the factor base?

I'm not talking about the size of the factor base, but which primes are candidates (only half are). Isn't there some easy way to test a prime to see if it can be?

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When using the quadratic sieve to factor $n$, the only primes you need in the factor base are the ones for which $n$ is a quadratic residue. Testing whether $n$ is a quadratic residue for a given prime modulus $p$ is easy (if $p$ is not too big) by using quadratic reciprocity and other properties of quadratic residues, which can be found in intro Number Theory texts.