0
$\begingroup$

Under Goldbach conjecture, denote by $ r_{0}(n) : =\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\} $ for any large enough composite integer $ n $ . Say a positive composite integer $ n $ is hexahedral if $ r_{0}(n)\mid 6 $. From Polymath8b, Elliott-Halberstam conjecture implies there are infinitely many pairs of primes $ (p,q) $ with $ q-p=12 $. Denoting for such pairs by $ n_{p,q} : =(p+q)/2 $, it follows that if the latter is composite, then $ r_{0}(n_{p,q})=6 $. If it is prime, then $ r_{0}(p+3)=3 $ . In any case, we get infinitely many hexahedral integers.

Now my question is : is the density of integers $ n $ such that $ r_{0}(n)=1 $ among hexaedral integers positive ? I think it should be equal to $1/6 $, which may be a consequence of Chebotarev's theorem.

Numerical experiments give a proportion of around $ 0.175... $ among hexaedral integers below $ 10^7 $ .

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.