p^q + q^p is prime

I found this elementary number theory problem in the “Problem Drive” section of Invariant Magazine (Issue 16, 2005), published by the Student Mathematical Society of the University of Oxford. Below, I have included the solution, which is very elementary.

Problem: Find all ordered pairs of prime numbers such that is also a prime.

Solution: Let . First, note that if is a solution, then so is . Now, and can’t be both even or both odd, else will be even. Without loss of generality, assume and some odd prime. So, . There are two cases to consider.

Case 1: .

This yields , which is prime. So, and, hence are solutions.

Case 2: .

There are two sub-cases to consider.

, where is some even integer. Then, we have . Hence, ; so, can’t be prime.

, where is some odd integer. Then we have . Hence, ; so, again, can’t be prime.

As we have exhausted all possible cases, we conclude and are the only possible solutions.

Inequality with log

This one, by Oleg Golberg, appeared in the issue of Mathematical Reflections (MR) 2007. I don’t have a solution yet, but I think I should be able to solve it sooner or later. If you find a solution, you should send it to MR by Jan 19. Here is the problem anyway.

For all integers , prove that

My first post

I became interested in mathematical blogging after visiting Terence Tao’s and Timothy Gower’s blogs on numerous occasions. It seems there is a sizable number of mathematicians disseminating valuable information through their blogs, and I see this as a healthy sign. Such blogs provide a wealth of information to students like me, and dare I say, I learn most of my math from such blogs!

I intend to write about math mostly as an exercise in exposition. I am assuming this will be of great help to me later on. I also will be posting some problems in the “Problem Corner” section every now and then.

Let’s see how this goes. I am hoping my enthusiasm for blogging will not wear off too soon!