[Note: there is an assignment from Class 8, due Thursday, Oct. 14. That is different from what is here, which is the assignment to do Thursday, October 14, and hand in Tuesday, Oct. 19.]
This class you are on your own! Feel free to work together, help each other out, etc., just write your answers to the TWO QUESTIONS independently. You will have to be in groups at the computers anyway, because there are more of you than there are machines. That's fine! You'll have to figure it out together. The assignment is to look through certain kinds of number patterns and see what you can find. Make use of software called "WINNUM," or (its MS-DOS equivalent) "NEWNUM," to be found in the Math 114 folder on all computers in the math department. You can find it by clicking the "Start" button at the lower left of the screen: a large menu appears, on which you should double-click the Math 114 folder. Then double click NEWNUM or WINNUM in that folder. The icon for WINNUM is green and yellow in a kind of speckly pattern, much like what appears when the program runs.
Both these programs simply display the positive integers up to some maximum, and highlight some subset of them. When the program starts, the primes are highlighted. The first problem is to make some sense of the pattern, or lack of it, formed by the primes.
The only freedom you have is to change the way the integers are displayed. When the program starts, the integers are written in an array of 10 columns: the first row is 0, 1, 2, ..., 9, the 2nd row is 10, 11, 12, ..., 19, etc. This means that the columns have a significance mod 10. The first column (0, 10, 20, ....) is 0 mod 10, the next column (1, 11, 21, ...) is 1 mod 10, etc. Even before you do anything, just look at the display. Where are the primes mod 10: i.e., which columns have the primes? (Some columns, like the one headed by 5, have no primes after the first, for a fairly obvious reason. I would say this is a column that is conspicuously MISSING primes. What you should notice is that there are NO MORE primes in that column.) Can you tell why the primes avoid certain columns?
Now change the display. Don't bother with the menu of the program! Just type "d" for "display." The program prompts you to enter the number of columns you want to display in, reminding you that you presently are displaying in 10 columns. You might try "7" and <ENTER>. Now the integers will be displayed in 7 columns, and you can see where the primes are mod 7 by seeing which columns have primes. Etc. Try many different ways of displaying the primes.
QUESTION 1: When are there primes which are k mod n? That is, when you display in n columns and look in column k (numbering from 0 as the first one), do you see a lot of primes or not? Answer for general n and k. What is the general rule??
The second exercise is aimed at questions like "Is there a number which is both 3 mod 11 and 5 mod 13?" Why don't we just answer that question using either of the programs WINNUM or NEWNUM: display in 11 columns with "d 11 <ENTER>". Now pick out the set which is 3 mod 11 by typing "c 3 <ENTER>". The command "c" means"column." The program prompts you for the number of the column you want. When you type 3 and <ENTER>, you highlight all the numbers which are 3 mod 11. Now display in 13 columns (with "d 13 <ENTER>") and look in column 5. The numbers there are, of course, the numbers which are 5 mod 13, but some of them are highlighted. These are numbers which are both 5 mod 13 and 3 mod 11. When I tried this I found solutions 135, 278, 421, ... .
Try another example: find a number which is both 7 mod 15 and 5 mod 9. If you try it, you will see there is no solution -- no number has that property. That peculiar result in this problem (given mod 15 and mod 9) never comes up in the previous problem (mod 13 and mod 11). Why not??
QUESTION 2: Experiment with problems of the form "Is there a number which is a mod m, and also b mod n?" When does such a number exist? When it does exist, what other numbers also solve the problem? Using fairly small numbers like 5, 6, 7 might be a good strategy here. Share your discoveries with others.
Have fun with these questions -- as you know, I am not looking for definitive solutions. I'm just hoping you will be creative and try things out, notice things, and even surprise yourself.