I’ve written up a draft version of a short paper giving the results we already have in the finding primes project.  The source files for the paper can be found here.

The paper is focused on what I think is our best partial result, namely that the prime counting polynomial has a circuit complexity of for some absolute constant whenever and .  As a corollary, we can compute the parity of the number of primes in the interval in time .

I’d be interested in hearing the other participants opinions about where to go next.  Ernie has suggested that experimenting with variants of the algorithm could make a good REU project, in which case we might try to wrap up the project with the partial result and pass the torch on.

The proposal “deterministic way to find primes” is not officially a polymath yet, but is beginning to acquire the features of one, as we have already had quite a bit of interesting ideas.  So perhaps it is time to open up the discussion thread a little earlier than anticipated.  There are a number of purposes to such a discussion thread, including but not restricted to:

1. To summarise the progress made so far, in a manner accessible to “casual” participants of the project.
2. To have “meta-discussions” about the direction of the project, and what can be done to make it run more smoothly. (Thus one can view this thread as a sort of “oversight panel” for the research thread.)
3. To ask questions about the tools and ideas used in the project (e.g. to clarify some point in analytic number theory or computational complexity of relevance to the project).  Don’t be shy; “dumb” questions can in fact be very valuable in regaining some perspective.
4. (Given that this is still a proposal) To evaluate the suitability of this proposal for an actual polymath, and decide what preparations might be useful before actually launching it.

To start the ball rolling, let me collect some of the observations accumulated as of July 28:

1. A number of potentially relevant conjectures in complexity theory and number theory have been identified: P=NP, P=BPP, P=promise-BPP, existence of PRG, existence of one-way functions, whether DTIME(2^n) has subexponential circuits, GRH, the Hardy-Littlewood prime tuples conjecture, the ABC conjecture, Cramer’s conjecture, discrete log in P, factoring in P.
   1. The problem is solved if one has P=NP, existence of PRG, or Cramer’s conjecture, so we may assume that these statements all fail.  The problem is probably also solved on P=promise-BPP, which is a bit stronger than P=BPP, but weaker than existence of PRG; we currently do not have a solution just assuming P=BPP, due to a difficulty getting enough of a gap in the success probabilities.
      1. Existence of PRG is assured if DTIME(2^n) does not have subexponential circuits (Impagliazzo-Wigderson), or if one has one-way functions (is there a precise statement to this effect?)
   2. Discrete log being in hard (or easy) may end up being a useless hypothesis, since one needs to find large primes before discrete logarithms even make sense.
2. If the problem is false, it implies (roughly speaking) that all large constructible numbers are composite.  Assuming factoring is in P, it implies the stronger fact that all large constructible numbers are smooth.  This seems unlikely (especially if one assumes ABC).
3. Besides adding various conjectures in complexity theory or number theory, we have found some other ways to make the problem easier:
   1. The trivial deterministic algorithm for finding k-bit primes takes exponentially long in k in the worst case.  Our goal is polynomial in k.  What about a partial result, such as exp(o(k))?
      1. An essentially equivalent variant: in time polynomial in k, we can find a prime with at least log k digits.  Our goal is k.  Can we find a prime with slightly more  than log k digits?
   2. The trivial probabilistic algorithm takes O(k^2) random bits; looks like we can cut this down to O(k).  Our goal is O(log k) (as one can iterate through these bits in polynomial time).  Can we do o(k)?
   3. Rather than find primes, what about finding almost primes?  Note that if factoring is in P, the two problems are basically equivalent.  There may also be other number theoretically interesting sets of numbers one could try here instead of primes.
4. At the scale log n, primes are assumed to resemble a Poisson process of intensity 1/log n (this can be formalised using a suitably uniform version of the prime tuples conjecture).  Cramer’s conjecture can be viewed as one extreme case of this principle.  Is there some way to use this conjectured Poisson structure in a way without requiring the full strength of Cramer’s conjecture?  (I believe there is also some work of Granville and Soundarajan tweaking Cramer’s prediction slightly, though only by a multiplicative constant if I recall correctly.)

See also the wiki page for this project.