620-113 Explorations in Number Theory


Course Outline

  • Outline,PDF (24 K)
  • Links

  • 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

  • Bibliography,PDF (58 K)
  • Lecture Notes

  • 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)
  • Assignment

    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)
  • Created: 22 January 2003
    Last modified: 26 July 2007
    Authorised by: Head, Department of Mathematics and Statistics.
    Maintained by: Nicholas Witte, Department of Mathematics and Statistics.
    Email: N.Witte@ms.unimelb.edu.au