Yours faithfully

Salau ayobami ]]>

I hope it is relevant.

]]>(the right-hand bound is due to Iwaniec; the left-hand side to Ford, Green, Konyagin, Maynard, and myself). The conjectures of Maier and Pomerance suggest should in fact grow like , which would imply that the proposition is false. It may be possible to investigate things numerically for small values of (for instance using the values in OEIS A048670), and perhaps even locate a counterexample. (Also, since you are only using shifts of the form , as opposed to all shifts between and , it may be possible to adapt the methods in my paper with Ford et al., or earlier and simpler arguments using the Erdos-Rankin method, to obtain an analytic disproof.)

Incidentally, it would be preferable if one could post mathematical questions as comments to blog posts whose topic is at least somewhat related to the content of the question (one can use the various search and categorisation options on the left sidebar to locate a suitable post, e.g., this one).

]]>Let the i-th prime number : .

Prouve that if and () then