Are even numbers the sum of two odd sub-primes? / How to use computers to check?

Question

Suppose $$p_{a} + p_{b} = 2{p_{1}}^{m_{1}}{p_{2}}^{m_{2}}...{p_{n}}^{m_{n}} \qquad \text{where }p_{x} \in \mathbb{P} -\left\{2\right\} $$

Then

$$\frac{p_{a} + p_{b}}{{p_{1}}^{m_{1}}{p_{2}}^{m_{2}}...{p_{n}}^{m_{n}}} = 2$$

Not including $2$; all prime numbers are odd. Therefore $p_{x}=2q_{x}+1$ so

$$\frac{2q_{a}+1 + 2q_{b}+1}{{p_{1}}^{m_{1}}{p_{2}}^{m_{2}}\dots{p_{n}}^{m_{n}}} = 2 = \frac{2(q_{a} + q_{b}+1)}{{p_{1}}^{m_{1}}{p_{2}}^{m_{2}}\dots{p_{n}}^{m_{n}}}$$

Therefore

$$q_{a} + q_{b}+1= {p_{1}}^{m_{1}}{p_{2}}^{m_{2}}\dots{p_{n}}^{m_{n}}$$

Therefore $q_{a} + q_{b}$ must be even because ${p_{1}}^{m_{1}}{p_{2}}^{m_{2}}\dots{p_{n}}^{m_{n}} - 1$ must be even.

For convenience we can call $q_{x}$ the sub-prime of the prime number $p_{x}$, where $q_{x} \in \mathbb{K}=\left\{1,2,3,5,6,8,9,11,14,15,18,20,21,23,26,29,...\right\}$ i.e $q_{x} = \frac{p_{x}-1}{2} : p_{x} \in \mathbb{P}-\left\{2\right\}$

$q_{a}$ and $q_{b}$ are either both odd or both even if the first supposition is true. Consequently it would follow that all even numbers are the sum of two sub-primes, but I ask, are all even numbers the sum of two ${\bf odd}$ sub-primes, are there any counter examples? (And how could I check this myself, where can I learn or what software could I use?)

Answer 1

First of all, it would be helpful if you explicitly defined a subprime to be a positive integer $q$ such that $2q+1$ is prime.

The problem of writing a positive integer $N$ (even or not) as the sum of two subprimes is equivalent, as you noted, to the problem of whether $2N+2$ is the sum of two primes. As far as any ideas we have on how to approach such problems, the subprime version is just as hard as the original Goldbach version.

So in short, we don't know how to establish the conjecture that every integer $\ge2$ is the sum of two subprimes.

Answer 2

Monitor[
  Module[{p, t}, 
    Do[Catch[For[p = Prime[t = 2], p < n, p = Prime[++t], 
      If[Mod[p, 4] == 3 && PrimeQ[n - p], Throw[p]]]; 
    Print[n]], {n, 6, 1*^8, 4}]],
  Refresh[n, UpdateInterval -> .2, TrackedSymbols -> {}]]

In response to your question about how you could check this question numerically, here is Mathematica code that I used to verify your conjecture up to $N=10^8$. The actual version of your conjecture I used is:

If $N\equiv2\pmod4$ and $N>2$, then $N$ is the sum of two primes $p,q\equiv3\pmod 4$.

It is not hard to see that this is equivalent to your formulation that every positive even number is the sum of two odd subprimes. Since the number theorists are still scratching their heads about Goldbach's conjecture, and your conjecture implies Goldbach's, this is probably the best you are going to get.