"Lecture" for Week 14

Tomorrow (Monday, Dec. 1) I will give a test to my other class, which must be returned the following Monday, but I will have very little time for grading during the week. Therefore, I will be grading all weekend, and there is no reason to ask you to submit your term papers this Friday. Let's move the date to next Monday (Dec. 8).

Meanwhile, I have not forgotten that you turned in book reviews and essays. I assure you that I have read all of them, and I'm basically pleased with them. Our course handout seems to mandate numerical grades on these things, which I hope to attach during this week. Again I urge you to put your reviews and essays on the Web if you haven't already.

This is my last weekly message, so let me say that it has been a pleasure to work with you. Although there were signs of stress over the last few weeks as the mathematical content (and perhaps other aspects of our lives) became more demanding, overall you have gone out of your way to let me know that you are enjoying and profiting from the course. Your enthusiasm sustained me through some occasions of self-doubt. I apologize for the slowness with which I've handled your written work. This semester has abounded in (often unexpected) interruptions and distractions, most of them professional but not course-related. I will spare you the details; I'm sure you hear enough whining from your students that you don't need it from your teacher. I'm still trying to figure out whether one can prove that e is transcendental without first knowing that pi is transcendental; then I'll grade that last homework! [In fact, the situation is really worse than that; see below.]

Algebra and number theory homework

Finally I looked at this. It's too late to post detailed solutions, but I'll try to make some brief, helpful comments.

• 2 (modular multiplication): Most of you already knew about this, but a few had trouble or made a mountain out of a molehill. Write both numbers in the form mq + r, do the multiplication, and write the result also in the form mq + r. Throw away all mq terms, making sure that your r terms are in the right range (0 through r - 1).
• 7 (Euler's formula): Expand the factors 1/(1 - 1/p) as geometric series. Contemplate multiplying the series together, and note that each denominator of the desired type occurs exactly once, with numerator 1. Working in the other direction (starting from the series of reciprocals) is harder!
• 13 (logarithmic series): The series given is the standard Taylor series for ln(1 - x). So either the x² on the left should just be x, or x everywhere on the right should be replaced by x².
• 16 (Gelfond => e is transcendental?): I finally got a chance to consult my department's number theorists, and so far they agree with Becky and me that the problem as written is impossible. The hypotheses of the Gelfond-Schneider theorem include that both the exponent and the base be algebraic. Therefore, the fact that the right-hand side (-1) is algebraic (and that we are not in one of the special cases, such as exponent being a rational real) tells us that either e or pi is transcendental, or both. There does not seem to be anything more to conclude without some other input. Even knowing that pi is transcendental (which was not proved until after the corresponding fact for e) does not help, since there is no contradiction in both being transcendental. So, one more problem to write off with an apology.

  News bulletin: Graduate student's computer finds largest known [Mersenne] prime number

  2^20,996,011 - 1 (6,320,430 decimal digits long)

  This news was in an AP story in the Bryan-CollegeStation Eagle, so you may have seen it too. For details, see http://www.mersenne.org and links thereupon.