CofC Math Student Solves Math Horizons Problem

By | January 21, 2014

Lauren Tubbs, a junior Mathematics Major at the College of Charleston, recently worked on solving a multi-part problem that was posed in the Playground section of the September 2013 issue of Math Horizons, a journal published by the Mathematical Association of America.  Lauren successfully solved Problem 295 on “Counting Divisors” and submitted her solution to the journal.  The problem statement is:

(a) Show that, if n is an odd number, then n^2+19 has at least six divisors, and that n^2+119 has at least eight divisors.
(b) Are 19 and 119 the best possible numbers we could have chosen for part a?  For this part, (still assuming n is an odd number), find the smallest positive integers a and b such that n^2+a has at least six divisors, and n^2+b has at least eight divisors.

Lauren’s solution is here for anyone who would like to read it.  A common challenge for undergraduate math students is simply: How do I get started on mathematical research?  Professor Dinesh Sarvate, who has directed many undergraduate research projects here at CofC, describes one way to get students started:

In general,  when a student approaches me for research or when I see that a student is  capable of some research,  I ask them to do such problems from Math Horizons, the American Mathematical Monthly, or the College Mathematics Journal before giving a more time consuming research problem. So I hope to get students involved for years to come.

Lauren describes her experience in getting started below:

Prof. Sarvate is my advisor and at my advising appointment he gave me some very good advice about which classes to take and about math in general. He also said that once I’d had discrete structures or abstract algebra, we could do a research project together. I was happy to take discrete structures I from him that summer, which I really liked, and in the fall as I was taking discrete structures II, I asked him if we could do a project. He gave me the Math Horizons problem to start with, making sure I understood exactly what the problem was asking. I was rather nervous at first and thought I wouldn’t be able to solve it as I’d never seen a similar problem before and had no idea how to tackle it. But it was an excellently chosen first problem because within an hour I had gotten the first two parts of the question and by that evening I had roughly figured out why the last two parts were true, though it took me a week to get a good proof and two weeks to write it up properly. The problem was also a great choice because it required almost no background: if it had been a geometry or calculus problem I would have first had to learn or relearn a lot of material, whereas with this number theory problem I was able to concentrate on the fairly new processes of problem-solving and proof-writing.

Lauren also describes how Professor Sarvate helped her in refining her solution to a manuscript she could submit to the journal:

I sent Prof. Sarvate my handwritten solution and he asked me to type it up formally in LaTeX. The first draft was something like four pages long. Prof. Sarvate went through the draft with a red pen and marked it up very carefully, crossing out whole paragraphs of unnecessary explanation and replacing clumsy phrases with more professional ones. This was extremely valuable since I had only ever written proofs for class before, and I was unsure what to include and how to phrase a solution for a journal. I really like this mathematics editing, trying to make things as elegant and concise as possible while remaining clear. With Prof. Sarvate’s help I eventually got the solution down to a single page. He also showed me how to format my solution and cover letter. It was submitted in October.

Overall, it was a great way to get introduced to problem solving and the research process, and Lauren also says that she hopes to work on a “real” research project in discrete mathematics when her course schedule allows for it.