Homework for Class 4

Of course, you may start on the homework during class if you have time!

1. Integer puzzle

   Recall the third-degree polynomial p you found in Class 2 having the values p(1)=2, p(2)=3, p(3)=5, p(4)=9. For instance, you could compute this polynomial in Maple via

          p:=n->a*n^3+b*n^2+c*n+d:
          subs(solve({p(1)=2,p(2)=3,p(3)=5,p(4)=9},{a,b,c,d}),p(n));
          p:=unapply(",n);
          

   Suppose an undergraduate student comes to you with the following question. ``I ran a Maple loop to compute the values of p at the integers from -10 to 10.''

          for i from -10 to 10 do
            p(i);
            od;
          

   ``I observed that these values are all integers, even though p has coefficients that are fractions! Can it be that p(n) is always an integer when n is, and if so, why?''

   Convince yourself that the answer is affirmative, and figure out a mathematical explanation. Then write, in LaTeX, an answer addressed to the student, and hand it in at the beginning of next class.

2. Internet scavenger hunt

   Use the Internet to find the following items. At the next class, hand in your answers together with the URLs where you found them.

   1. The World-Wide Web home page of the town where you were born. If there is no such home page, find the home page of a city as close as possible to the place where you were born. (Hint: check out City.Net.)

   2. The title of a mathematical publication by a person (not yourself or a member of your immediate family) whose name differs from yours by as few letters as possible. For example, the mathematician Harald Bohr has a name that is a close match to mine.

   3. A computer-generated fractal image.

   4. A picture of the Borromean rings.

   5. The probabilities of the different hands in standard five-card poker.

   6. The title of (a) my senior thesis and (b) my PhD thesis.

   7. Information about the largest known prime number. (A new one was discovered in early September 1996.)

   8. The URL of an electronic version of Alice's Adventures in Wonderland; also the title of a serious mathematical work by the same author.

   9. The schedule for the ICTCM conference to be held in Reno, Nevada in November of this year.

Comments to Harold P. Boas.
Created Sep 24, 1996. Last modified Sep 25, 1996.