There are no integer solutions at all !

For the proof, assume the contrary that there are x, y such that

3*(x + y) + 12*(x^2 + y^2) + 16*(x^3 + y^3) = 2138156388.

Then the remainders on each side after dividing each by n will also be the same for ALL integers n. That is

3*(x + y) + 12*(x^2 + y^2) + 16*(x^3 + y^3) = 2138156388. (mod n)

After a bit of trial and error, we let n = 7 and test for x, y = 0 ... 6

Now 2138156388 = 4 but 3*(x + y) + 12*(x^2 + y^2) + 16*(x^3 + y^3) can only equal {0,1,3,5,6}.

Hence the assumption that there are integer solutions is false.

The method also works here for n = 14, 21 ,...