The polymath blog

August 28, 2009

(Research Thread IV) Determinstic way to find primes

Filed under: finding primes,research — Terence Tao @ 1:43 am
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This post will be somewhat abridged due to my traveling schedule.

The previous research thread for the “finding primes” project is now getting quite full, so I am opening up a fresh thread to continue the project.

Currently we are up against the “square root barrier”: the fastest time we know of to find a k-digit prime is about $\sqrt{10^k}$ (up to $\exp(o(k))$ factors), even in the presence of a factoring oracle (though, thanks to a method of Odlyzko, we no longer need the Riemann hypothesis).  We also have a “generic prime” razor that has eliminated (or severely limited) a number of potential approaches.

One promising approach, though, proceeds by transforming the “finding primes” problem into a “counting primes” problem.  If we can compute prime counting function $\pi(x)$ in substantially less than $\sqrt{x}$ time, then we have beaten the square root barrier.

Currently we have a way to compute the parity (least significant bit) of $\pi(x)$ in time $x^{1/2+o(1)}$, and there is hope to improve this (especially given the progress on the toy problem of counting square-frees less than x).  There are some variants that also look promising, for instance to work in polynomial extensions of finite fields (in the spirit of the AKS algorithm) and to look at residues of $\pi(x)$ in other moduli, e.g. $\pi(x) mod 3$, though currently we can’t break the $x^{2/3+o(1)}$ barrier for that particular problem.

August 13, 2009

(Research Thread III) Determinstic way to find primes

Filed under: finding primes,research — Terence Tao @ 5:10 pm
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This is a continuation of Research Thread II of the “Finding primes” polymath project, which is now full.  It seems that we are facing particular difficulty breaching the square root barrier, in particular the following problems remain open:

1. Can we deterministically find a prime of size at least n in $o(\sqrt{n})$ time (assuming hypotheses such as RH)?  Assume one has access to a factoring oracle.
2. Can we deterministically find a prime of size at least n in $O(n^{1/2+o(1)})$ time unconditionally (in particular, without RH)? Assume one has access to a factoring oracle.

We are still in the process of weighing several competing strategies to solve these and related problems.  Some of these have been effectively eliminated, but we have a number of still viable strategies, which I will attempt to list below.  (The list may be incomplete, and of course totally new strategies may emerge also.  Please feel free to elaborate or extend the above list in the comments.)

Strategy A: Find a short interval [x,x+y] such that $\pi(x+y) - \pi(x) > 0$, where $\pi(x)$ is the number of primes less than x, by using information about the zeroes $\rho$ of the Riemann zeta function.

Comment: it may help to assume a Siegel zero (or, at the other extreme, to assume RH).

Strategy B: Assume that an interval [n,n+a] consists entirely of u-smooth numbers (i.e. no prime factors greater than u) and somehow arrive at a contradiction.  (To break the square root barrier, we need $a = o(\sqrt{u})$, and to stop the factoring oracle from being ridiculously overpowered, n should be subexponential size in u.)

Comment: in this scenario, we will have n/p close to an integer for many primes between $\sqrt{u}$ and u, and n/p far from an integer for all primes larger than u.

Strategy C: Solve the following toy problem: given n and u, what is the distance to the closest integer to n which contains a factor comparable to u (e.g. in [u,2u])?  [Ideally, we want a prime factor here, but even the problem of getting an integer factor is not fully understood yet.]  Beating $\sqrt{u}$ here is analogous to breaking the square root barrier in the primes problem.

1. The trivial bound is u/2 – just move to the nearest multiple of u to n.  This bound can be attained for really large n, e.g. $n =(2u)! + u/2$.  But it seems we can do better for small n.
2. For $u \leq n \leq 2u$, one trivially does not have to move at all.
3. For $2u \leq n \leq u^2$, one has an upper bound of $O(n/u)$, by noting that having a factor comparable to u is equivalent to having a factor comparable to n/u.
4. For $n \sim u^2$, one has an upper bound of $O(\sqrt{u})$, by taking $x^2$ to be the first square larger than n, $y^2$ to be the closest square to $x^2-n$, and noting that $(x-y)(x+y)$ has a factor comparable to u and is within $O(\sqrt{u})$ of n.  (This paper improves this bound to $O(u^{0.4})$ conditional on a strong exponential sum estimate.)
5. For n=poly(u), it may be possible to take a dynamical systems approach, writing n base u and incrementing or decrementing u and hope for some equidistribution.   Some sort of “smart” modification of u may also be effective.
6. There is a large paper by Ford devoted to this sort of question.

Strategy D. Find special sequences of integers that are known to have special types of prime factors, or are known to have unusually high densities of primes.

Comment. There are only a handful of explicitly computable sparse sequences that are known unconditionally to capture infinitely many primes.

Strategy E. Find efficient deterministic algorithms for finding various types of “pseudoprimes” – numbers which obey some of the properties of being prime, e.g. $2^{n-1}=1 \hbox{ mod } n$.  (For this discussion, we will consider primes as a special case of pseudoprimes.)

Comment. For the specific problem of solving $2^{n-1}=1 \hbox{ mod } n$ there is an elementary observation that if n obeys this property, then $2^n-1$ does also, which solves this particular problem; but this does not indicate how to, for instance, have $2^{n-1}=1 \hbox{ mod } n$ and $3^{n-1}=1 \hbox{ mod } n$ obeyed simultaneously.

As always, oversight of this research thread is conducted at the discussion thread, and any references and detailed computations should be placed at the wiki.

August 9, 2009

(Research thread II) Deterministic way to find primes

Filed under: finding primes,research — Terence Tao @ 3:57 am
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This thread marks the formal launch of “Finding primes” as the massively collaborative research project Polymath4, and now supersedes the proposal thread for this project as the official “research” thread for this project, which has now become rather lengthy. (Simultaneously with this research thread, we also have the discussion thread to oversee the research thread and to provide a forum for casual participants, and also the wiki page to store all the settled knowledge and accumulated insights gained from the project to date.) See also this list of general polymath rules.

The basic problem we are studying here can be stated in a number of equivalent forms:

Problem 1. (Finding primes) Find a deterministic algorithm which, when given an integer k, is guaranteed to locate a prime of at least k digits in length in as quick a time as possible (ideally, in time polynomial in k, i.e. after $O(k^{O(1)})$ steps).

Problem 2. (Finding primes, alternate version) Find a deterministic algorithm which, after running for k steps, is guaranteed to locate as large a prime as possible (ideally, with a polynomial number of digits, i.e. at least $k^c$ digits for some $c>0$.)

To make the problem easier, we will assume the existence of a primality oracle, which can test whether any given number is prime in O(1) time, as well as a factoring oracle, which will provide all the factors of a given number in O(1) time. (Note that the latter supersedes the former.) The primality oracle can be provided essentially for free, due to polynomial-time deterministic primality algorithms such as the AKS primality test; the factoring oracle is somewhat more expensive (there are deterministic factoring algorithms, such as the quadratic sieve, which are suspected to be subexponential in running time, but no polynomial-time algorithm is known), but seems to simplify the problem substantially.

The problem comes in at least three forms: a strong form, a weak form, and a very weak form.

1. Strong form: Deterministically find a prime of at least k digits in poly(k) time.
2. Weak form: Deterministically find a prime of at least k digits in $\exp(o(k))$ time, or equivalently find a prime larger than $k^C$ in time O(k) for any fixed constant C.
3. Very weak form: Deterministically find a prime of at least k digits in significantly less than $(10^k)^{1/2}$ time, or equivalently find a prime significantly larger than $k^2$ in time O(k).

The pr0blem in all of these forms remain open, even assuming a factoring oracle and strong number-theoretic hypotheses such as GRH. One of the main difficulties is that we are seeking a deterministic guarantee that the algorithm works in all cases, which is very different from a heuristic argument that the algorithm “should” work in “most” cases. (Note that there are already several efficient probabilistic or heuristic prime generation algorithms in the literature, e.g. this one, which already suffice for all practical purposes; the question here is purely theoretical.) In other words, rather than working in some sort of “average-case” environment where probabilistic heuristics are expected to be valid, one should instead imagine a “Murphy’s law” or “worst-case” scenario in which the primes are situated in a “maximally unfriendly” manner. The trick is to ensure that the algorithm remains efficient and successful even in the worst-case scenario.

Below the fold, we will give some partial results, and some promising avenues of attack to explore. Anyone is welcome to comment on these strategies, and to propose new ones. (If you want to participate in a more “casual” manner, you can ask questions on the discussion thread for this project.)

Also, if anything from the previous thread that you feel is relevant has been missed in the text below, please feel free to recall it in the comments to this thread.

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