Is the conjecture that every integer $n \equiv 2 \bmod{4}$ greater than $6$ can be expressed as the sum of two primes of the form $4n+3$ a strengthening, or a consequence of Goldbach?
2026-03-25 19:03:11.1774465391
Strengthening or consequence of Goldbach?
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It is neither a strengthening nor a consequence.
A number congruent $2$ modulo $4$ could also be the sum of two primes that are congruent to $1$ modulo $4$. Your condition excludes this latter possibility. So it cannot follow from Goldbach.
Your conjecture says nothing about number congruent $0$ modulo $4$ and there is no way (at least no apparent one) to get them as sums from your assertion.
However, your conjecture should be at a similar level of difficulty as the Goldbach conjecture; likely slightly harder.
Note though that every number congruent $6$ modulo $8$ is a sum of two primes congruent to $3$ modulo $4$ would be a consequence of Goldbach's conjecture, as a representation of such a number as a sum of two primes cannot involve a prime congruent $1$ modulo $4$.