Outline,PDF (24 K)
The Prime Pages (Chris Caldwell)
The World of Mathematics (Eric Weisstein)
Mersenne Primes (GIMPS team)
A Prime Case of Chaos (Barry Cipra)
Closing the Gap on Twin Primes (Ivars Peterson)
Small gaps between prime numbers: the work of Goldston-Pintz-Yildrim (K. Soundararajan)
Progressive Primes (Ivars Peterson)
Bibliography,PDF (58 K)
Preliminary Notes,PDF (30 K)
Lecture 1,PDF (60 K)
Lecture 2,PDF (64 K)
Lecture 3,PDF (83 K)
Lecture 4,PDF (50 K)
The assignment will be given out after the second lecture, and the last date for submission is Thursday 24th August. It is preferable that you complete and hand them after we have discussed the material in class rather than leaving them to the last minute as this gives me time to give you some feedback. Place assignments in the 620-113 box (number 1) in the North corridor of the Richard Berry Building or bring them to class. Don't forget to complete and sign a plagiarism cover-sheet, available from the Department Office, and attach it with your assignment work.
Assignment,PDF (59 K)
The project part of the Assignment involves guessing an approximate equation for the number of primes less
than a given integer for large values of this number
and is best done by plotting the data. The data set can be downloaded from the link
Data set of first 1000 primes
or the links to the larger data sets. You may even want to generate more than
100,000 primes using your own code but this is entirely optional.
If you don't have plotting software on your own computer or can't access this on
a University machine then plotting software for MS Windows, OS2, Mac and Linux
platforms is available without cost at
Gnuplot Version 4.0.
This is a very comprehensive plotting package for scientific purposes. You can also
download the software from the following link
FTP repository for Gnuplot,
but be patient for this page to load. Once you see the file directory then look for
the compressed package, the .zip file appropriate for your system. If you have a
machine running MS Windows for example then download the gp400win32.zip file.
Save it to your file system and unpack it using WinZip (which seems to be included
in standard systems). It will be unpacked into
a subdirectory \gnuplot and in the \bin subdirectory of this will be the
executable file pgnuplot.exe. It is this binary that you need to run. Documentation
on gnuplot is also unpacked and you may like to look at gnuplot.pdf.
Start up gnuplot and use the following commands to analyse your data
set style data steps
set size square
pi(x)=a function of x using binary operations *,+,-,/ and common functions log(x),exp(x) etc
plot [x=2:50] [0:40] pi(x),'1000primes.txt'
In the 3rd command is where you define your guess function and you can use the up
arrow on your keyboard to recover previous function definitions, edit them and try them
out again. The 4th command is the plot command and it assumes that you have the data file
in the same directory as the Gnuplot binary. You can adjust the x and y ranges to see
how well you are going in different ranges.
Once you have found a good guess then you can examine the error or difference
between the guess and the data with the following command
plot [x=2:7919] [-5:12] '1000primes.txt' using ($1):($2)-pi($1)
This is a more exacting test of your guess so you may need to go back to one of
the earlier steps and try something new or a little bit more sophisticated -
remember that we want a fairly simple equation.
Once you have got a good fit then you can print out the graphs using the following
series of commands, which assumes one has a postscript printer but use the specific
device that you have (there is a list in the documentation)
set output 'output.ps'
set terminal postscript
then issue the plot command again, which will send the plot to the file output.ps
which you can print out. To have the plot displayed in a window again just use
set terminal win
If you get into strife then come and see me. We can even try things out on one of the
PCs in the Computer Lab.
Maple worksheets, Data and PS files
Maple worksheet of simple examples,mw (136 K)
Sieve of Eratosthenes (up to 499),PDF (11 K)
First 1000 primes,PDF (15 K)
Data set of first 1,000 primes,TXT (13 K)
Data set of first 10,000 primes,TXT (105 K)
Data set of first 100,000 primes,TXT (1240 K)