I want to collect a list of Goldbach's other conjectures

Question

I want to collect a list of Goldbach's other conjectures. I konw only two conjectures: The first one is the famous statement on writting an even number as the sum of two primes and the other one is about expressing an even number by $3$ primes.

Answer 1

I don't know which conjecture is that about expressing an even number by $3$ primes, but Goldbach also conjectured, in a letter to Euler written in 1752, that every odd number can be written as $2n^2+p$, with $p$ prime. Euler checked it for every number up to $2\,500$. It turns out that the conjecture is false: in 1856, Moritz A. Stern, a professor of mathematics at Göttingen, found two numbers which could not be written as twice a square plus a prime, namely $5\,777$ and $5\,993$. These seem to be the only known counter-examples to this conjecture.

Answer 2

Actually, there is another "Goldbach conjecture" which Goldbach was able to prove himself: in $1752$, Goldbach showed that no polynomial with integer coefficients can give a prime for all integer values (Nagell 1951, p. 65; Hardy and Wright 1979, pp. 18 and 22).

Answer 3

I am turning @GerryMyerson's comment into an answer because it is useful and relevant to the question.

---

You can find all of Goldbach's correspondence with Euler at http://eulerarchive.maa.org//correspondence/correspondents/Goldbach.html (but it's all in Latin).

From the Publication Information on the website:

167 letters from the Euler-Goldbach Correspondence were published by P.H. Fuss in his Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle . The letters below all came from Fuss' book. Unfortunately, Fuss often excised the personal part of the correspondence, and published only the scientific material. A complete set of the correspondence will have to wait until the publication of the appropriate volume of the Opera Omnia