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

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.

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.
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).