Introduce yourself

You are encouraged to use the comments to this post to introduce yourself to the approximately 60 or so new Rowan University freshman math majors and to the faculty in the math department.

Wolfram|Alpha and Mathematica

In May 2009 Wolfram|Alpha was released. It describes itself as a computational knowledge engine. Among the many things it does, its mathematical abilities are very useful. It can function like an incredibly souped up graphing calculator (although comparing it to a graphing calculator is like comparing Google to a paper telephone directory). Queries can be written informally such as "sin(x^2) from -2 to 3" or "integral of x+log(x)" (note that it will even show all steps of the computation) or "10th prime number". Not only does it do such basic computations, it also has an immense library of much more advanced abilities in higher mathematics, statistics, and applied sciences. It even provides easy access to real numerical data. Since it is free and works anywhere on any modern internet browser I expect that it will revolutionize both the teaching and practice of mathematics at all levels. I encourage you to spend some time playing with Wolfram|Alpha. It is an exciting time to be doing mathematics.

Wolfram|Alpha takes short one line queries. For more detailed programming, Rowan University has a student license for Mathematica. Once you have your student account and email, Rowan University students can download Mathematica for Students from Rowan's password protected website www.rowan.edu/download for Macs or Windows. Once installed on your own computer you will have to register and get your personal password using your Rowan email address as instructed. Mathematica is used in many math courses at Rowan. Wolfram|Alpha can save its output in Mathematica format and formal Mathematica syntax can be used in Wolfram|Alpha.

Some remarkable math

It is best to start with some good math.

It is true that 2^p-2 is always a whole multiple of p for any prime number p. For example if p=5 then 2⁵-2 = 32-2 = 30 = 6 times 5 which is a multiple of 5. Similarly if p=7 then 2⁷-2 = 128-2 = 126 = 18 times 7 which is a multiple of 7. However if we take the nonprime p=6 we get 2⁶-2 = 64-2 = 62 which is not a multiple of 6.

Question 1: Does this work if we replace 2 by other integers such as 3? (In which case we would we would be asking whether 3^p-3 is a multiple of p.)

Questions 2: We know that the result does not hold for the nonprime p=6. But could there be any nonprime p for which this works?