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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?

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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.

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  • $\begingroup$ "No" is a singularly unhelpful response. Some guidance as to where others have in fact considered this reformulation, and what they had to say about it, would be much more enlightening. $\endgroup$ – Keith Backman Feb 28 '13 at 20:33
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    $\begingroup$ @KeithBackman, This algebraic manipulation is trivial and works regardless of whether or not p,q are prime. For these reasons it brings nothing to the table so you won't find it mentioned in literature or studied. I disagree that "no" is unhelpful, I normally refrain from commenting on questions which have negative answers (e.g. here the answer is obviously "no" but..) since no one likes negative answers - but I felt it would be of value to you base on your previous question. $\endgroup$ – user58512 Feb 28 '13 at 20:44
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    $\begingroup$ More generally: no reformulation of Goldbach, twin-prime, Collatz, etc., etc., that fits into a paragraph of high school algebra is going to open any avenues of analysis. That's no reason to stop thinking about these problems, but it is a reason to expect a very curt reply from those familiar with the history. $\endgroup$ – Gerry Myerson Mar 1 '13 at 3:02
  • $\begingroup$ @user58512 Why do you say the question you mentioned has an obvious answer of "no"? For one, it seems like too complicated a question to be answered by a simple "yes" or "no." Second of all, the answers so far have been quite interesting and have noted that there might be some connections (although not necessarily what the OP thought). I'd be interested to see your answer as to why there's "clearly" no relationship. $\endgroup$ – Stahl Mar 2 '13 at 0:17
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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.

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