consider non-primitive congruences, but these congruences only contain finitely many primes and are thus of less interest to solving the Erdős-Straus conjecture (and if the Erdős-Straus conjecture held for a common factor of $r$ and $q$, then the residue class $r \mod q$ would trivially be solvable by polynomials.

By Dirichlet’s theorem, the primitive residue class $r \mod q$ contains arbitrarily large primes $p$. For each large prime $p$ in this class, we either have one or two of the $P_1(p), P_2(p), P_3(p)$ divisible by $p$, as observed previously. For $p$ large enough, note that $P_i(p)$ can only be divisible by $p$ if there is no constant term in $P_i$. We thus conclude that either one or two of the $P_i(n)$ have no constant term, but not all three. Let us call the congruence *Type I solvable* if one can take exactly one of $P_1,P_2,P_3$ to have no constant term, and *Type II solvable* if exactly two have no constant term. Thus every solvable primitive residue class $r \mod q$ is either Type I or Type II solvable.

It is well-known (see [@Oblath:1950; @mordell]) that any primitive residue class $n = r \mod 840$ is solvable by polynomials unless $r$ 