Goldbach for Hurwitz Primes

Having a Goldbach problem for positive integers, for positive Gaussian integers, it is natural formulate one for Hurwitz integers. Here is a conjecture:

Every Lipschitz Quaternion integer (a,b,c,d) with a>1,b>1,c>1,d>1 is the sum of two Hurwitz primes (a,b,c,d) + (1,1,1,1)/2, with a>0,b>0,c>0 and d>0.


A Lipschitz integer is the ring of quaternions (a,b,c,d), where a,b,c,d are all integers. A Hurwitz integer is the ring of quaternions of the form (a,b,c,d) or (a+1/2,b+1/2,c+1/2,d+1/2). A Hurwitz prime z is a Hurwitz integer which can not be written as a product pq of two other Hurwitz integers p,q where both have norm smaller than the norm of z. Due to the non-commutativity of the quaternions, there is no unique prime factorization but since the Euclidean algorithm still works and since quaternions satisfy N(z w) = N(z) N(w) forming a division algebra, one can still talk about primes.

Since the above conjecture has never been formulated, we don't know how hard it is yet but we can relate it to some Bunyakovsky type problems. The computer shows that it is true for smaller integers. Here is a special case: Let us look at Lipschitz integers of the form (a,b,c,d) = (2,2,2,n). Now there are two ways to write this as a sum of two primes (up to permutations). Either (1,1,1,2k+1)/2 + (3,3,3,2n-2k-1)/2 or (1,1,3,2k+1)/2 + (3,3,1,2n-2k-1)/2. This means that the Hurwitz Goldbach conjecture implies that for every integer n, we can find a k smaller than n such that either
          1+k+k^2     and      7 - (n-k) + (n-k)^2    are prime
    
or then that
          3+k+k^2     and      5 - (n-k) + (n-k)^2    are prime
    


The Hurwitz Goldbach conjecture implies that there are either infinitely many primes of the form 1+k+k^2 or infinitely many primes of the form 3+k+k^2.


Proof: If not, then there would be a maximal C such that for m=n-k>C both 8-m+m^2 and 5-m+m^2 were always prime. But this fails for example for m being a multiple of 5 already. Therefore, the Goldbach conjecture is harder than a Landau type problem. The conjecture is natural since it tells that for the function f(x,y,z,w) = sum(a,b,c,d) Hurwitz prime xa yb zc wd that f(x,y,z,w)2 has positive derivatives dk/dxk dl/dyl dm/dzm dn/dwn f(x,y,z,w)2 for all k,l,m,n bigger than 1. It is a calculus problem for a concrete function of four variables. Note that x,y,z,w are usual variables and not quaternions. The question can be asked for any Beuerling type set of primes in the quaternions.