## Posts Tagged ‘polymath5’

### 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…)