# Math 124: Fall 2002 Final Examination

## Due Sunday 12 January 2003 at 5pm

### Call me at 617-308-0144 or email [email protected] in case of questions.

 DVI file: final.dvi Postscript file: final.ps PDF file: final.pdf HTML version: latex2html The LaTeX source file: final.tex, macros.tex

## Questions and Answers About the Exam

• > For 12 b, can we show that x^3+ax+b has a repeated root instead,
> because that's what we need in c to show that f has nonzero
> discriminant.

Yes, but remark that it is well-known that a cubic has a repeated root
if and only if it has zero discriminant.

• On Saturday 11 January 2003 06:11 pm, you wrote:
> perfect square.  The problem is that MAGMA doesn't have the precision
> to compute ceiling(sqrt(n)).  Is there a way to do this?

Check out

http://magma.maths.usyd.edu.au/magma/htmlhelp/text565.htm#4679

• > In 5a, a and b are to be integers, right?

Yes.

• > In 5b, does the sum of the divisors include 1? On the P.S. with a
> similar question, that function included 1 and n, but in this case
> obviously doesn't include n.

sigma is the sum of the proper divisors, so sigma(1)=1, sigma(2)=1,
sigma(3)=2, sigma(6)=1+2+3.

• > For this, you want some rational number a/b, a,b integers, such that
> a/b expanded in decimal is equal to the given floating point up to the
> accuracy given, not absolutely equal, right?

Answer the problem as stated using the rules at the beginning of the
exam.  If you think it is impossible to find such a rational number,
prove it to get full credit.

• > I'm working on problem 5, part b.  It's clear that I can use the
> algorithm that I devised in the homework assignment a few months ago in
> which we have sigma(n) and phi(n) and we want to factor it into pqr.
> But I've done everything in the exact same way that I did before, using
> MAGMA, and when I factor the polynomial to get the factorization, I get
> 2, some really big negative number with a nontrivial decimal, and 1.
> This is really weird. I'm just trying to check and see that there's not
> a typo in the number or anything?

I'm not going to tell you whether or not there is a typo.  However, if
you can carefully prove that there must be a typo in the number then
you will get full credit for the problem.

• > One more thing...if we do more than 8 problems, will you take the best
> 8 solutions, or will you just ignore the additional ones (i.e. should I
> try to do a couple extra if I have time)?

Turn in solutions to exactly eight problems.   If you turn in more than
eight, I will grade the first eight lowest numbered problems.  E.g., if
you turn in 1-7, 9, 10, and 12, I will discard 10 and 12.

• > For problem 3 ("characterize the numbers n such that Z/n is a field",
> etc.) do we have to prove our answers?  I assume so, but just checking.

Yes, but for this problem you can be very terse and cite results from
the course notes.

• > I'm trying to factor these numbers in number 5, and I see you've
> changed the ECM intrinsic that you used in class to perform ECM on
> large numbers.  I tried "ECM;" but I don't understand what step ratio
> and number of steps arguments should be in order for the function to be
> effective (if it even is effective?) on the number in 5a.  Can you
> explain what these arguments are?

No comment... except that we discussed many factoring methods besides
ECM.

• > on p. 12.2 on the final, should it be 4a^3+27b^2=0 or 4a^3-27b^2=0?

It's technically a typo, but it doesn't make any difference since
if 4a^3-27b^2 = 0 then -(4(-a)^3+27b^2)=0 so 4(-a)^3 + 27b^2 = 0.
I.e., there is a bijection between the (a,b) for 4a^3+27b^2=0 and
for 4a^3-27b^2=0 given by (a,b) <--> (-a,b).

> in p. 7, where are x_0, y_0, \alpha_1, etc. supposed to live?

I'll let you decide; make the choice that is most interesting to a
mathematician.  For example, in 7(a) if the alpha_i, beta_i are only
required to lie in the complex numbers then the statement is trivial,
hence not interesting.