I need some help with this algorithm

Question

I'm quite new to Mathematica yet, so there's probably a lot of not-so-subtle problems (indeed the algorithm isn't even running yet). My goal with this algorithm is to input an integer $n$ and get all Goldbach partitions of all even integers > 4 up to $2n$. I'd like it to be as efficient as possible, since this is part of a bigger code, and I really want all partitions for each even. What I considered: I'm gonna input an integer $n$, and I'll return a list (of lists of 2-uples) $G$ with all partitions, wherein $G$[[0]] will give the list of all partitions of the first even > 4 and so on. Also, for efficiency sake, I'm willing to calculate only the primes between $2(n-1)$ and $2n$ each iteration, but I guess that's far from clear on what's coded there. This justifies, though, the prime list $Pr$, since I'm just appending up the new primes that happen at each iteration. So, the algorithm is to calculate the next prime (which is the 'primepi[2t]-th' prime, put it on the list, calculate the sum of all 2-combinations of the list of primes and check whether it sums up to $2t$; if so, it'll append this tuple to $G$.

 GTest[n_Integer] := 
    Pr = List[];  G = List[]; k = 0;
    ParallelTable[
    Append[Pr, Prime[PrimePi[2*t]]] && 
    Do[( k = Subsets[Pr, {2}][[j, 0]] + 
    Subsets[Pr, {2}][[j, 1]] ) && 
   (If[k == 2*t, Append[G, Subsets[Pr, {2}][[j]]]]),
   {j, 0, Length[Subsets[Pr, {2}] - 1]}],  {t, 3, 2*n}]

The error I'm getting is quite long, I'll paste just the ten last lines:

(kernel 1) Append::normal : Nonatomic expression expected at position 1 in Append[Int,366].

(kernel 2) Part::partd : Part specification {}[[0,1]] is longer than depth of object.

(kernel 1) Part::partd : Part specification {}[[0,1]] is longer than depth of object.

(kernel 2) Append::normal : Nonatomic expression expected at position 1 in Append[Int,330].

(kernel 1) Append::normal : Nonatomic expression expected at position 1 in Append[Int,368].

(kernel 2) General::stop : Further output of Append::normal will be suppressed during this calculation.

(kernel 1) General::stop : Further output of Append::normal will be suppressed during this calculation.

(kernel 2) Part::partd : Part specification {}[[0,1]] is longer than depth of object.

(kernel 1) Part::partd : Part specification {}[[0,1]] is longer than depth of object.

(kernel 2) General::stop : Further output of Part::partd will be suppressed during this calculation.

(kernel 1) General::stop : Further output of Part::partd will be suppressed during this calculation.

During evaluation of In[53]:= Append::normal: Nonatomic expression expected at position 1 in Append[Int,6].

During evaluation of In[53]:= Append::normal: Nonatomic expression expected at position 1 in Append[Int,8].

During evaluation of In[53]:= Append::normal: Nonatomic expression expected at position 1 in Append[Int,10].

During evaluation of In[53]:= General::stop: Further output of Append::normal will be suppressed during this calculation.

Answer 1

You are indeed quite new to Mathematica. I would recommend learning that arrays are indexed 1-based rather than 0-based before leaping straight into a parallel table. And you can't picks two things out of a list of size one. And P is not the same as Pr. But enough admonishment. I'd just use the built in partitioning to do it.

ClearAll@GoldbachPartitions;
Attributes[GoldbachPartitions] = {Listable};
GoldbachPartitions[n_] := IntegerPartitions[2 n, {2}, Prime@Range@PrimePi[2 n]]

In action:

GoldbachPartitions@123

{{241, 5}, {239, 7}, {233, 13}, {229, 17}, {227, 19}, {223, 23}, {199, 47}, {193, 53}, {179, 67}, {173, 73}, {167, 79}, {163, 83}, {157, 89}, {149, 97}, {139, 107}, {137, 109}}

GoldbachPartitions@Range[2, 10]

{{{2, 2}}, {{3, 3}}, {{5, 3}}, {{7, 3}, {5, 5}}, {{7, 5}}, {{11, 3}, {7, 7}}, {{13, 3}, {11, 5}}, {{13, 5}, {11, 7}}, {{17, 3}, {13, 7}}}