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Let $E$ be the smallest even number that cannot be expressed as the sum of two primes. By definition, $E\neq p+3$. But it is also the case that $E-2\neq p+3$ and $E-4\neq p+3$, because if otherwise, then $E$ could be expressed as $p+5$ or $p+7$, violating the assumption. (Numbers $E$ such that $E\neq p+3$, $E-2\neq p+3$, and $E-4\neq p+3$ are listed in OEIS A283555.)

Since $E$ is the smallest even number that cannot be expressed as the sum of two primes, $E-2$ and $E-4$ must each have one or more pairs of prime addends ($p$ and $q$). Since no such $p$ or $q$ can be 3, all must be of the form $6k\pm 1$.

If $E$ cannot be expressed as the sum of two primes, then at a minimum neither prime addend summing to $E-4$ can be supplemented by 4 to give a prime, and neither prime addend summing to $E-2$ can be supplemented by 2 to give a prime.

The first condition is necessarily met if both prime addends summing to $E-4$ both have the form $6k-1$, implying $E-4\equiv -2 (mod 6)$, hence $E\equiv 2 (mod 6)$. The second condition is necessarily met if both prime addends summing to $E-2$ have the form $6k+1$, implying $E-2\equiv 2 (mod 6)$, hence $E\equiv -2 (mod 6)$. But it is not possible that $E\equiv 2 (mod 6)$ and $E\equiv -2 (mod 6)$, so both conditions cannot be simultaneously true. Whichever condition fails, it is then possible that some pair of prime addends of either $E-2$ or $E-4$ will give rise (by supplementation of one of the pair by either 2 or 4) to a pair of primes summing to $E$, thus invalidating the assumption.

My question is, can anybody suggest ways in which the stated possibility can be strengthened to a proof?

To clarify that question, we can see that assuming $E$ to be the smallest even number to fail the Goldbach test, the possibility must exist for the some prime addends of $E-2$ and $E-4$ to contradict the assumption. Can it be shown that among those prime addends, there must be a case where at least one such addend can be supplemented by 2 or 4 (as appropriate) to yield a prime, for example by proving that if no set of prime addends of $E-2$ can be supplemented by 2 to yield a prime, then it would have to be the case that at least one of the prime addends of $E-4$ would necessarily be supplementable by 4 to yield a prime, thus in essence proving Goldbach?

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    $\begingroup$ The first thing you may want to consider is breaking this block of text into some paragraphs, so it becomes more readable. $\endgroup$ – Bram28 Mar 12 '17 at 21:08
  • $\begingroup$ Yes you can something about $E,p,q \bmod 6$. Is it your question ? $\endgroup$ – reuns Mar 12 '17 at 21:10
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    $\begingroup$ If elementary considerations were going to lead to a proof of Goldbach, Euler would have found it. $\endgroup$ – Gerry Myerson Mar 13 '17 at 6:11

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