# The polymath blog

## June 29, 2010

### Draft version of polymath4 paper

Filed under: discussion,finding primes — Terence Tao @ 8:36 pm

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 $\sum_{a < p < b} t^p \hbox{ mod } 2$ has a circuit complexity of $O(x^{1/2-c+o(1)})$ for some absolute constant $c>0$ whenever $1 < a < b < x$ and $b-a < x^{1/2+c}$.  As a corollary, we can compute the parity of the number of primes in the interval $[a,b]$ in time $O(x^{1/2-c+o(1)})$.

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.

## June 12, 2010

### Mini-polymath proposal: IMO 2010 Q6

Filed under: news,planning,polymath proposals — Terence Tao @ 11:16 pm

I am proposing the sixth question for the 2010 International Mathematical Olympiad (traditionally, the trickiest of the six problems) as a mini-polymath project for next month.  Details and discussions are in this post on my other blog.

[Update, June 27: the project is scheduled to start on Thursday, July 8 16:00 UTC.]

## January 9, 2010

### Polymath5: Erdős’s discrepancy problem

Filed under: polymath proposals — Gil Kalai @ 10:56 pm

Is taking place on Gowers’s blog!

## December 31, 2009

### Proposal (Tim Gowers): Erdos’ Discrepancy Problem

Filed under: polymath proposals — Gil Kalai @ 3:11 pm

For a description of Erdos’ discrepancy problem and a large discussion see this blog on Gowers’s blog.

The decision for the next polymath project over Gowers’s blog will be between three projects: The polynomial DHJ problem, Littlewood problem, and the Erdos discrepency problem. To help making the decision four polls are in place!

## November 20, 2009

### Proposals (Tim Gowers): Polynomial DHJ, and Littlewood’s problem

Filed under: polymath proposals — Gil Kalai @ 10:06 am

Tim Gowers described two additional proposed polymath projects. One about the first unknown cases of the polynomial Density Hales Jewett problem. Another about the Littelwood’s conjecture.

I will state one problem from each of these posts:

1) (Related to polynomial DHJ) Suppose you have a family $\cal F$ of graphs on n labelled vertices, so that we do not two graphs in the family $G,H$ such that $H$ is a subgraph of $G$ and the edges of $G$ which are not in $H$ form a clique. (A complete graph on 2 or more vertices.) can we conclude that $|{\cal F}|$ =$2^{o({{n} \choose {2}}}$? (In other words, can we conclude that $\cal F$ contains only a diminishing fraction of all graphs?)

Define the “distance” between two points in the unit cube as the product of the absolute value of the differences in the three coordinates. (See Tim’s remark below.)

2) (Related to Littlewood) Is it possible to find n points in the unit cube $[0,1]^3$ so that the “distance” between any two of them is at least $1/100000000000000000000n$?

A negative answer to Littlewood’s problem will imply a positive answer to problem 2 (with some constant). So the pessimistic saddle thought would be that the answer to Problem 2 is yes without any bearing on Littlewood’s problem.

## November 8, 2009

### Proposal (Tim Gowers) The Origin of Life

Filed under: polymath proposals — Gil Kalai @ 12:54 pm

A presentation of one possible near future polymath: the mathematics of the origin of life can be found on Gowers’s blog.

## October 27, 2009

### (Research thread V) Determinstic way to find primes

Filed under: finding primes,research — Terence Tao @ 10:25 pm

It’s probably time to refresh the previous thread for the “finding primes” project, and to summarise the current state of affairs.

The current goal is to find a deterministic way to locate a prime in an interval $[z,2z]$ in time that breaks the “square root barrier” of $\sqrt(z)$ (or more precisely, $z^{1/2+o(1)}$).  Currently, we have two ways to reach that barrier:

1. Assuming the Riemann hypothesis, the largest prime gap in $[z,2z]$ is of size $z^{1/2+o(1)}$.  So one can simply test consecutive numbers for primality until one gets a hit (using, say, the AKS algorithm, any number of size z can be tested for primality in time $z^{o(1)}$.
2. The second method is due to Odlyzko, and does not require the Riemann hypothesis.  There is a contour integration formula that allows one to write the prime counting function $\pi(z)$ up to error $z^{1+o(1)}/T$ in terms of an integral involving the Riemann zeta function over an interval of length $O(T)$, for any $1 \leq T \leq z$.  The latter integral can be computed to the required accuracy in time about $z^{o(1)} T$.  With this and a binary search it is not difficult to locate an interval of width $z^{1+o(1)}/T$ that is guaranteed to contain a prime in time $z^{o(1)} T$.  Optimising by choosing $T = z^{1/2}$ and using a sieve (or by testing the elements for primality one by one), one can then locate that prime in time $z^{1/2+o(1)}$.

Currently we have one promising approach to break the square root barrier, based on the polynomial method, but while individual components of this approach fall underneath the square root barrier, we have not yet been able to get the whole thing below (or even matching) the square root.  I will sketch the approach (as far as I understand it) below; right now we are needing some shortcuts (e.g. FFT, fast matrix multiplication, that sort of thing) that can cut the run time further.

## October 16, 2009

### Nature article on Polymath

Filed under: news — Terence Tao @ 4:25 pm

Timothy Gowers and Michael Nielsen have written an article “Massively collaborative mathematics“, focusing primarily on the first Polymath project, for the October issue of Nature.

## August 28, 2009

### (Research Thread IV) Determinstic way to find primes

Filed under: finding primes,research — Terence Tao @ 1:43 am
Tags:

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
Tags:

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.

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