Stronger Goldbach-related conjecture

[rokirovka]

For any positive integer n>1, define gap(n) as

gap(n) = |n-p| = |n-q| for p,q the two primes closest to n such that p+q = 2n.

The Goldbach Conjecture is equivalent to the existence of p and q, and thus the existence of gap(n), for all n>1. (Or all n>2 if p and q are required to be odd primes, which does not affect the substance of the conjecture.)

But I make the following conjecture, which is stronger than the mere existence of gap(n) for all n>1:

For all n>1, the maximum value of gap(n)/n is gap(22)/22 = 9/22 = .40909... (for primes p=13 and q=31).

Other high values of gap(n)/n are

gap(8)/8 = 3/8 = .375 (p=5, q=11)
gap(46)/46 = 15/46 = .326... (p=31, q=61)
gap(28)/28 = 9/28 = .321... (p=19, q=37)
gap(32)/32 = 9/32 = .28125 (p=23, q=41)
gap(58)/58 = 15/58 = .259... (p=43, q=73)
gap(4)/4 = 1/4 = .25 (p=3, q=5)

Other n for which gap(n)/n > .2 include [in descending order of gap(n)/n] n=49, 25, 38, 146 [gap(146)=33 for p=113, q=179], 9, 68, 55, 14, 24, 74.

 

[dodo]

Hello, rokirovka,
I don't know much about the subject, but I would imagine that a weaker version of your conjecture, like

If gap(n) exists, then it is not bigger than 9n/22​

would be more useful, in terms of bounding the solution space.

 

[rokirovka]

And surely it is much more likely that a large gap(n)/n exists for some n, than it is that gap(n) does not exist at all for some n.

The former is a lack of primes p,q that sum to 2n within a certain large range from about 3n/5 to 7n/5. The latter is a lack of primes p,q that sum to 2n within the entire range from 1 to 2n.

So if your weaker version of the conjecture is true, that would imply with overwhelming likelihood the truth of the stronger version as well. Of course "overwhelming likelihood" is not an acceptable form of mathematical proof. But proving your weaker version could be the right way to begin to tackle the problem.