Comments on: (Research thread II) Deterministic way to find primes

By: Phillip Brix

Phillip Brix — Tue, 06 Mar 2012 18:51:40 +0000

one (dumb) approach to solving problem 1 and 2, assuming you can detect whether or not a number is prime fairly quickly, would be the following.
start off with a binary number c bits long, containing all 1′s. test for prime. change the lowest bit to 0. test for prime. change the second lowest bit to 0. test for prime. change the lowest bit back to 1, test for prime. continue changing 1 bit at a time and testing for prime. in the worst case, it will take 2^(c/2) changes to find a prime. for problem 1, this means that you can find a prime of size log_2(k) in k time. for problem 2, this means you can find the largest prime of size log_2(k) in k steps.

By: Mathematics, Science, and Blogs « Combinatorics and more

Mathematics, Science, and Blogs « Combinatorics and more — Sat, 09 Jan 2010 23:06:46 +0000

[...] skeptical and enthusiastic. (August 2009) polymath4 dedicated to finding primes deterministically was launched over the polymathblog. (Polymath4 was very active for several months. It led to some fruitful [...]

By: Four Derandomization Problems « Combinatorics and more

Four Derandomization Problems « Combinatorics and more — Sun, 06 Dec 2009 09:30:05 +0000

[...] Polymath4 is devoted to a question about derandomization: To find a deterministic polynomial time algorithm for finding a k-digit prime. So I (belatedly) devote this post to derandomization and, in particular, the following four problems. [...]

By: Polymath again « What Is Research?

Polymath again « What Is Research? — Tue, 27 Oct 2009 23:38:35 +0000

[...] meantime, Terence Tao started a polymath blog here, where he initiated four discussion threads (1, 2, 3 and 4) on deterministic ways to find primes (I’m not quite sure how that’s [...]

By: Gil

Gil — Fri, 14 Aug 2009 06:14:05 +0000

Just a little comment regarding the factoring oracle/additive combinatorics approach. What we did was to look at an interval of length k^5 of k digits numbers; and to conjecture that some number in the interval has a large prime factor. (Where “large” can be a number with at least m digits for m= k/log k or even just m=k^{1/3} numbers.)

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.

By: Polymath4 « Euclidean Ramsey Theory

Polymath4 « Euclidean Ramsey Theory — Thu, 13 Aug 2009 23:54:21 +0000

[...] http://polymathprojects.org/2009/08/09/research-thread-ii-deterministic-way-to-find-primes/ [...]

By: Gil Kalai

Gil Kalai — Thu, 13 Aug 2009 21:44:40 +0000

Thanks! I suppose we can ask our pronlem abstractly for Beurling primes where we have cost 1 for finding the nth “integer”, testing “primality” and (perhaps) “factoring”.

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.

By: Terence Tao

Terence Tao — Thu, 13 Aug 2009 17:12:10 +0000

As this thread is now quite full, I am opening a new research thread for this project at

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

By: (Research Thread III) Determinstic way to find primes « The polymath blog

(Research Thread III) Determinstic way to find primes « The polymath blog — Thu, 13 Aug 2009 17:10:50 +0000

[...] finding primes, research — Terence Tao @ 5:10 pm Tags: polymath4 This is a continuation of Research Thread II of the “Finding primes” polymath project, which is now full. It seems that we are [...]

By: Terence Tao

Terence Tao — Thu, 13 Aug 2009 16:41:04 +0000

If one uses the “base u” approach then the fact that n is small is reflected in the fact that there are only a few digits in the base u expansion of n, which keeps the relevant dynamical system approximately “polynomial” of low degree. If n is less than u^2, then one has a linear dynamical system not too dissimilar from the rotation by example. What this system tells us is that if , then there is an integer within of n which has a factor near u, but this is trivial since having a factor near u is the same as having a factor near n/u.