Reformulating the Goldbach conjecture in a quasi-Pythagorean form Goldbach conjectured that every sufficiently large even number $2n$ can be expressed as the sum of two primes, i.e. $2n=p+q$.  I will assume for this question that $p$ and $q$ are distinct odd primes, so 'sufficiently large' means $n \geq 4$.  An equivalent statement of Goldbach is that every integer $n \geq 4$ is the average of two distinct prime numbers, $n=(p+q)/2$.  If we assume $p>q$ and define $k=(p-q)/2$, we see that $p=n+k$ and $q=n-k$.  Multiplying, we get $pq=n^2-k^2$.  Rearranging, we arrive at a reformulation of Goldbach, namely: every integer $n$ has a square that can be expressed as the sum of a smaller square plus an odd non-square semiprime: $n^2 = k^2 + pq$.  This equation bears a superficial resemblance to the Pythagorean equation.   Since $pq$ is odd, $n$ and $k$ must have opposite parity, and unlike primitive Pythagorean solutions, $n$ can be even (a quick examination shows that the assumption that $p$ and $q$ are distinct implies all solutions to the present equation must be primitive).  This reformulation turns Goldbach from a question about sums to a question about products.  My question is, does such a reformulation open other avenues of analysis, perhaps even proof, of Goldbach?
 A: No.
Not knowing how to rewrite $$2n = p+q$$ to $$n^2 = \left(\frac{p-q}{2}\right)^2 + pq$$ is not the obstruction that has blocked people from proving the Goldbach conjecture for the last three hundred years.
A: Long Comment
I think this particular reformulation of Goldbach's Conjecture is of interest to those amateurs wishing to think about the underlying connections between problems in elementary number theory. 
As an example take $n=22$. Only considering odd integers greater than 2, we have 5 potential solutions {p,q} where $2n=p+q$, they are {3,19}, {5,17}, {7,15}, {9,13} and {11,11}. Only three of these are correct solutions, consisting of prime $p$ and $q$; that is {3,19}, {5,17} and {11,11}.
If we look for solutions in the form 
$$n^2-\left(\frac{p-q}{2} \right)^2=pq$$
The restatement of the Goldbach conjecture would be something like:
Consider the finite set of numbers for any given $n$ which are each equal to the difference of two squares $n^2 - b^2$ where n and b are integer and $n\ge4$ and $(n-2)>b>2$. If (the modern formulation of) the Goldbach conjecture is true then for all $n\ge4$ there must be least one number in this set of numbers $n^2 - b^2$ that is semi-prime (i.e. the product of only two primes), all the other numbers in the set being higher composite numbers (i.e. the product of three or more primes). 
Therefore in this reformulation we need to study the distribution of semi-prime numbers relative to higher composite numbers, within an infinite number of finite sets of the form described above. 
In the standard version of the Goldbach Conjecture the distribution of primes below $2n$ for all $n$ needs to be studied directly. 
It is not at all obvious to me that this is a trivial restatement of the conjecture. Do we know enough about the distribution of semi-primes relative to higher composites to say that it is? 
If there are good references in the math. literature showing why this is a trivial restatement then this would be very useful valuable response to Keith Blackman's question.
A: Every odd number can be written as the difference of two perfect squares. Thus, every odd (not necessarily but for matters of this topic square-free) semiprime can be written as the difference of two perfect squares. Then, if one proves that there is at least one odd square-free semiprime between any two adjacent perfect squares that can be written by subtraction of a perfect square from the higher adjacent perfect square, this should also prove Goldbach's conjecture. - Example for the odd square-free semiprime 33:  33 + 16 = 49 with 3 + 11 = 7 + 7 = 14, example for the odd square-free semiprimes 39 and 55:  39 + 25 = 55 + 9 = 64 with 3 + 13 = 5 + 11 = 8 + 8. - This way, Goldbach's conjecture can be rewritten as a statement about the distribution of odd square-free semiprimes between consecutive perfect squares.
