A weaker conjecture that will be as fine for us is this: if we start with an interval of length k^5 and consider the set S of all m-digit or more factors (not necessarily primes) of all the numbers in the intervals; then there will be an m-digit prime factor for a number in S+S.

In other words, we can allow several interations of the operations: “taking sums” and “taking factors”.

The razor is as bad to this idea as it was for the original one. But if we can somehow find a way around the razor, this extension can be helpful because the sum/product theorems suggest that you can do more if you allow more than one arithmetic operation.

I remembered vagely a proof by Renyi to PNT which used very little about primes and it may be related to Beurling result. I also remember vaguely that even if you take every prime with probability 1/2, conjectures for primes beyond the PNT fails.
My question was in trying to suggest how to save additive NT methods against the razor.

The “Liphshitz” or “no large gaps” condition looks very strong. Even if you delete m primes you create by the Chinese remainder theorem a gap of length m near their product. But our razor which was based on deleting all large primes involved in an interval of length k^3 (say) of k-digits integers amounts to creating unusually large gaps. So there is a (small) hope that some additive number theory methods (or other methods) can exploit somehow a “no gaps” condition or a “no unusually large gaps” condition.

http://polymathprojects.org/2009/08/13/research-thread-iii-determinstic-way-to-find-primes/

http://www.ams.org/mathscinet-getitem?mr=2218342

though my university does not have a subscription to this year of the journal.

The reviewer does however notes an elementary way to attain the square root bound here, or more precisely to show that given any u (not necessarily integer), there is an integer in with a factor in . Indeed, one lets be the first integer larger than u, thus ; then one lets be the closest square number to , thus and , as claimed. (Note that this improves upon the Gauss circle problem type argument I had in #7, where I needed an interval of size or so.)

If one could somehow adapt this argument to primes instead of integers, we could attain the square root barrier without using RH.

In the above paper, apparently the square root barrier is broken assuming some strong exponential sum bound relating to the equidistribution of modulo 1 (this seems reminiscent of the dynamical systems approach).