Project
(Draft version)

One of the important skills that you will need to develop, both in the academic and the non-acdemic world, is the ability to present technical material to a variety of audiences.  This project will give you the opportunity to prepare and give a presentation to an audience of your peers.

Each of you will be expected to read a section (or more) of the textbook, do a little more research on that topic, and present that material in a lecture format to the class.  You will also be expected to suggest (at least four) homework problems (chosen from the even problems in the text) and to write up solutions to those problems.

Some interesting topics include:

• Round robin tournaments (5.3)
  
• Check digits and coding theory (5.5)
  
• Perfect Numbers and Mersenne primes (7.3)
• Cryptography (7, but ask about specific material)
• Pseudorandom Numbers (10.1)
• ElGamal (10.2)
• Zero-Knowledge proofs (11.5)
  

You should hand in:

• A copy of your class notes
• Detailed summaries of what you read outside of the textbook
• Solutions to homework exercises that you tried (at least 6, two or more involving proofs)

You will be graded on:
 

• The quality of your presentation
• How well you seemed to understand the material, using the written material you provide
  
• How much you researched the material
• The quality of your work on the problems
  

Problems:

• Section 7.3: 4, 10
• Prove that ((sigma(p^a))/p^a) < 1 + 1/(p-1)
• Prove that ((sigma(p^a))/p^a) >= p/(p-1)  - 1/((p-1)p)
• Section 5.5, #20, 12
• Suppose  we use the following scheme for finding a check digit.  The code word is an 11-digit number with (base 10) digits; the last digit is the check digit.  This check digit is calculated by looking at the first 10 digits as if they formed an actual 10-digit number (i.e. the word 1000000000 would be the number 10^9) and figuring out what that number is mod 9.  (In the example, 1000000000 is congruent to 1 mod 9.)  Show that the only transposition error that gets detected is when the check digit is transposed with another digit.  (Note you  need to show two things: this error does get detected and  no other transposition error is detected.
• Section 5.3, #2
• One way to represent a round robin tournament is by using a N-1 x N matrix.  The entry in row k and column i is the team that i plays against in round k.  Table 5.1 in section 5.3 is an example of this kind of matrix, as is the first example that Guy gave.  Make a similar table for the circle-chord method when N=6.  Compare this table to the table in example 5-14-1.
• Suppose N is an even, positive integer bigger than 2.  Suppose you construct two round robin tournaments, one by using the method that Rosen describes and one by using the circle-chord method with the numbers from 1 to N-1 arranged in order around the circle.  Show that the resulting tables (see previous problem) have the same rows but in different order.  (in other words, show that there is a bijection from the rows of one table to rows of the other.)  (ask me if you're confused.)
• Section 8.5, #2, 8, 10, 12