Midterm 1

[Math 115 Student]

Hello,
Will the fist midterm cover everything covered in class up to and
including the lecture on Friday 9/22? The 9/22 lecture hasn't been
covered on a homework yet, so this is why I ask.
Thanks.

[Kenneth A. Ribet]

> Will the fist midterm cover everything covered in class up to and
> including the lecture on Friday 9/22? The 9/22 lecture hasn't been
> covered on a homework yet, so this is why I ask.

Yes, in principle it covers everything through last Friday. The
lecture yesterday (Monday, September 25) is not in the scope of the
exam. On Friday, I explained Don Zagier's proof that primes that are
1 mod 4 can be written as the sum of two squares. That proof is not
in the book, so there are no handy HW exercises for it that I can
assign.

Good luck to all!

Ken R

[John Brooks-Jung]

Speaking of homework, have any solutions to them been posted anywhere?
If not will they be in the future. I am interested in the one about
the inductive proof of the four number theorem as my answer was
lacking.

Thanks,
John Brooks-Jung

[Kenneth A. Ribet]

> Speaking of homework, have any solutions to them been posted anywhere?
> If not will they be in the future. I am interested in the one about
> the inductive proof of the four number theorem as my answer was
> lacking.

I have decided not to post them. If you post solutions, they get
indexed by search engines, and then anyone in the world taking a
course from the book can look for HW answers on the web. A standard
work-around is to post solutions on a password-protected web page.
My alternative work-around was to e-mail solutions to all registered
students using the campus's courseweb system. The e-mail goes to
each student's official e-mail address. In some cases, this might be
some berkeley.edu address that the student never checks any more. If
you're in this situation, try to change your official campus e-mail
address to reflect the account that you actually check. (If not,
maybe you can forward from your berkeley.edu address to one that you
like more.)

The solutions have been written by Aaron G., our GSI. Thanks, Aaron!

In the particular problem that you mention, the strategy is supposed
to work by induction on b.
You start with ab = cd and try to get r, s, t and u by applying the 4-
number theorem to a second quadruple (a,b,c,d) in which b is
smaller. The idea is to divide b into d and concoct a new quadruple
in which the role of b is plated by the remainder in the division.
There's a tiny wrinkle that the write-up doesn't mention. Namely, in
the case where b divides d exactly, the remainder is 0 instead of a
positive integer; on the other hand, the a, b, c and d are probably
intended to be positive. If b does divide d exactly, the best
strategy is probably not to appeal to the inductive hypothesis but
instead to write down r, s, t and u directly. I did this in my
office today for a couple of students, and the choices for r, s, t
and u were pretty much forced on me.

Hope this helps,
Ken R