=========================================== Mass mailing to all registered students 18 OCT AM
Folks,
As previously announced (and described on the course webpage) there is a written homework assignment due Monday Oct 20th in lieu of the test originally scheduled for that day.  The next announced test for Monday NOV 3 will take place, and be a test.

David Phillips has called my attention to a certain vagueness concerning #3 in the homework due Oct 20th.  A PRIME NUMBER is a positive integer, evenly disible only by itself and 1.  One major theorem--usually called the "Fundamental Theorem of Arithmetic"-- in number theory is that every positive integer is uniquely factorizable into prime numbers.  Still another important theorem is that there are infinitely many prime numbers.  (This is not immediately obvious, since they get sparser as you get larger: there are five between 1 and 10, 1,2,3,5,7, but only two between 20 and 30 between 23,29, and so on.  All prime numbers are odd.)

The prime factors are the result of this factorization, and are the positive prime factors other than 1.

We will say that 12 has THREE prime factors (2,2, and 3) even though it has only two distinguishable prime numbers in its factorization ({2,3}).
Likewise, 4 has two prime factors (2 and 2) and so on.

This is what was not perfectly clear in the assignment.

If you need more information, you might read the short, understandable discussion within the long article on number theory in Encyclopedia Britannica (logon through the UB Library website).
You needn't be fearful since there are only a few details of number theory that seem to be important for logic (another is the Chinese Remainder Theorem); they all deal with primes and factorizability.
If you have any questions, feel free to write me over the weekend.
Randy Dipert   


=======================================Response to Question on #3    18 OCT PM
I actually worked one exactly like this in the class that dealt with diagonalization; it might be in your (or somebody's) notes.

Delta-L is not a number but a SET of numbers.  To see if 1 is in Delta L, one looks in S(1); to see if 2 is in Delta L, one looks in S(2), and so on.  One counts, starting with N=1 and applies the all-important condition:

N is IN Delta L IFF N is NOT in S(N).

Thus, since I only give you S(1) through S(14) you can check to see if each of the numbers 1-14 are in, or not in, Delta L.  The answer could theoretically be a set of numbers (all between 1 and 14) as large as 14 and as small as the empty set.