Archive for January, 2010

Correlations of the first 1124 sequence

January 25, 2010

Here is some data on the partial sums $PS_k(m):=\sum_{i=1}^mx_ix_{i+k}$ (with the convention $x_{i+k}=0$ when $i+k>m$) for small values of $k$. (more…)

Installing Xcode and using GCC on Mac

January 18, 2010

[Update (december 2011): the blog post below was initially written in january 2010. I guess it can still be quite useful to complete beginners or people who have a not-so-recent Mac, but please note the following.  First, the new version 10.7 of Mac OS X called Lion has been released in july 2011, and can be bought and downloaded in the App Store for \$29.99 if you have the latest Snow Leopard .    Second, a new generation of Xcode, which runs only on Lion has been released, called X code 4.  As I write it is Xcode 4.2.1 and it can be downloaded for free from the App Store.  It is only with the Xcode 4 generation that you can developp Apps for iPad and iPhone: have a look at the iOS dev center, while if you want to developp stuff for Mac you should look at the Mac dev center.]

Introduction: This tutorial has been written for the mathematician who knows very little about programming yet who wishes to use a given source code as a “black-box”, that is to produce an executable program from this code, and run that program on the Mac using one’s own data files.

I have thus made the tutorial a very explicit step-by-step one with screenshots.   The aim is  to install on a Mac and use the GCC compiler, which allows to create executable programs from a C or C++ source code.   (For Windows and linux versions click here).

On a Mac, GCC comes in particular as part of a nice package called Xcode, which is a full IDE (Integrated Development Environment: a software that makes writing and managing code a simpler process).  So we shall see in detail how to install Xcode, and will use only a very tiny part of it to compile our source code.

Average over quadruples of the first 1124 sequence

January 15, 2010

Here is a plot corresponding to the computation for the first 1124 sequence of the average of $x_ax_bx_cx_d$ over all quadruples $a,b,c,d \leq N$ such that $ab=cd$ (Note: to cut the computation time I’ve only taken those with $a and $c, this  without loss of generality, thanks to the commutativity of the quadruple product which doesn’t care about the order and to the additivity of the average)  (click to enlarge).

More data on multiplicative growth

January 15, 2010

Now I’ve made some new plots using Tim’s exact proposition 4 in the wish list:  choose either -1 or +1 at prime p depending on which minimizes the discrepancy up to the next prime q.

Some first data on growth of D in the multiplicative case

January 14, 2010

I’ve now put on the wiki a C++ program that allows to investigate the growth of D as a function of length for completely-multiplicative sequences, depending on how one chooses the values at prime indices.

Number of multiplicative sequences and plot (C=2)

January 13, 2010

Using Alec’s python script I’ve made the following plots of the number of multiplicative sequences with C=2.

The maximum length is 246.   The plots look quite structureless… (more…)

Super-quick tutorial for polymath5 users

January 12, 2010

Here’s a super quick tutorial on how to compile C++ code and use it as a black box.  I’m starting with Mac following Tim’s request.

Note: this post is likely to evolve quite a lot, I’ll try to set up a nicer tutorial over the next days, having in mind a mathematician who wants to go straight at the point.

Number of sequences as a function of length

January 11, 2010

Here are the number of valid sequences of short length satisfying C=2.

Could one cook up an explicit (possibly recursive) formula? How does that compare to the Erathosthene sieve: can we see a valid sequence as an analog of a prime number, and if so can we adapt a proof (say Euclid’s) that there are infinitely many primes to this context?
(more…)

A sequence of 584 elements with T2(x)=-x

January 10, 2010

Here are tables of Alec’s 584 sequence satisfying the constraint $x_{2n}=-x_n$.
(more…)

Tables for a C10 candidate

January 10, 2010

Here are tables for a $C_{10}$-symmetry candidate sequence provided by Alec.
(more…)