This material is due on Tuesday, December 22 (the day of the nonexistent official Math 458 exam). You can submit your work electronically, or by snail mail to me at 10 Chester Street, Newton, MA 02461.

If enough of you decide to complete your work by Thursday, December 17, we'll meet that day at school to go over some of the problems, and celebrate. (Those who haven't finished shouldn't come.)

1. From Lily's presentation:

   1. Are Lily's integral quaternions the only ones with integral norms?

   2. Show that the integral quaternions are closed under multiplication.

   3. Show that the norm is multiplicative.

   4. Write each of 31, 103 and 3193 as a sum of four squares. Please try to use Lily's theorems rather than brute force. Can you find multiple essentially different solutions in any of these cases?

   5. Find an infinite set of integers that cannot be written as a sum of three squares.

2. From Deanna's presentation: (page 1, page 2, her paper (pdf), and the Pinasco paper she reported on)

   1. Let p₁, ..., p_n be distinct odd primes and N their product. Show that the asymptotic density of the set of integers divisible by at least one of the p_i is 1 - phi(N)/N. Explain why this is intuitively clear.

   2. Is the assertion in the previous problem correct if the primes are not distinct?

   3. Let E be the set of positive integers with an even number of digits. Show that E does not have an asymptotic density. (That's really too bad, since intuition says that a random integer should have an even number of digits with probability 1/2.)

   4. Does the argument you made in the previous problem work for any base?

3. From Matt's presentation: (Here are his slides. You'll need OpenOffice to look at them.

   1. Suppose m and n are positive integers with m not equal to n. Can the integers 1,2,...,mn be inserted into an m x n rectangle in such a way as to make the sums of the entries of all the rows and columns the same? If not, can you at least make all the row sums the same and all the column sums the same?

   2. Can you fill an nxn square grid with the numbers 1,...,n² such that every row and column has the same product rather than sum?

   3. Can you fill an nxn square grid with n² distinct integers such that every row and column has the same product rather than sum?

   4. Produce a filled, magic, diabolic 3x3 square with the integers 1,...,9, or show that no such square exists.

      Reminder: A square is diabolic if the broken diagonals as well as the main diagonals add to the magic sum.

   5. An nxn Latin square is a square grid filled so that each row and each column contains each of the numbers 0,...,n-1. Two Latin squares are orthogonal if when they are superimposed each of the n² pairs (x,y) appears in some cell.

      Show that if you have a pair of orthogonal Latin squares you can construct a magic square by mapping the pair (x,y) to the number with digits x and y in base n.

      Find an example of a magic square constructed this way.

      Can every magic square be constructed this way? (I just invented the question and don't know the answer, nor do I know whether the question is hard or easy. I do have a conjecture.)

      What happens if you think of (x,y) as representing the complex number x+iy?

   6. Prove that if A and B are magic squares of the same size then so is AB (the matrix product).

      For a hint, work an example and guess how to compute the magic sum of the product from the magic sums of the factors.

      This is easy linear algebra - you should not have to write an ugly proof with summations and indices. I didn't think I knew it until Matt discovered it and suggested this problem. When I thought about it some more I realized I'd known it for a long time, in disguise.

4. From Koffi and Peter's presentation: (slides, text)

   1. Show that the continued fraction expansion of a rational number a/b is finite. Relate the arithmetic operations you performed to those needed to compute the greatest common divisor of a and b using the Euclidean algorithm.

   2. Write C_k = A_k/B_k for the k^th convergent to the continued fraction

      [a0, a1, a2, ... ]
      

      Prove that

      AkBk-1 - Ak-1Bk = (-1)k
      

      (I may have the indices off by 1.)

   3. Compute the continued fraction expansion of sqrt(58) and use it to find the fundamental solutions of the Pell equations

      	x2 - 58 y2 = -1
      

      and

      	x2 - 58 y2 = 1.
      

      Then find another solution to each of these equations.

   4. (Optional - please get this right if you decide to do it.) Prove that all the solutions to the Pell equation

      	x2 - d y2 = 1	
      

      come from powers of the first solution (x,y) for which alpha = x+sqrt(d)*y is greater than 1.

      Hint: Divide any solution beta by the largest power of alpha that's less than beta.

      Extra optional problem (for Peter and Koffi?) - prove that alpha is in fact the fundamental solution to the Pell equation.

5. From Joe and Shane's presentation:

   1. Since 11 == 1 (mod 5), the Legendre symbol (11/5)=1. The argument Joe and Shane presented shows that then (5,11)=1 by finding an explicit square root of 5 (mod 11). Work through their argument numerically to find that square root.

   2. Let p be an odd prime. Show that if p == 2 (mod 3) then every number has a cube root mod p, while if p == 1 (mod 3) then exactly one third of the nonzero residues have cube roots mod p.

      Hint: Let g be a primitive root for p. What powers of g have cube roots (mod p)?

   3. Let p and q be odd primes. Show that if p == 1 (mod q) then exactly 1/q of the nonzero residues of p have q^th roots mod p. (When q = 3 this is just the previous problem.)

      Note: this question does not enter directly into the proof Joe and Shane presented. But it does motivate a piece of it. Can you see where?

6. About Ethan's presentation:

   Math 458 this semester has been an experiment for me. I've never taught it this way before. I enjoyed it. I'd appreciate knowing whether (and if so how) you did. Please take a few minutes to write down some reflections. Here are some questions you might consider in your short essay:

   • What did you enjoy most?
   • What least?
   • What might I do differently next time?
   • Did you learn more or less (or differently) than in a more traditionally structured course?
   • Did this course take more (or less) of your time than a traditionally structured course?
   • What do you imagine you will remember? What will you probably forget?

   Finally, and completely optional: what grade do you think you've earned? (I may or may not use your answer to this question in determining your grade.)