  We formulate some refinements of Goldbach's conjectures based on heuristic
arguments and numerical data. For instance, any even number greater than 4 is
conjectured to be a sum of two primes with one prime being 3 mod 4. In general,
for fixed $m$ and $a, b$ coprime to $m$, any positive even $n \equiv a + b
\bmod m$ outside of a finite exceptional set is expected to be a sum of two
primes $p$ and $q$ with $p \equiv a \bmod m$, $q \equiv b \bmod m$. We make
conjectures about the growth of these exceptional sets.
