Another elementary number theory problem

This one, by Dr. Titu Andreescu (of USAMO fame), is elementary in the sense that the solution to the problem doesn’t require anything more than arguments involving parity and congruences. I have the solution with me but I won’t post it on my blog until Jan 19, 2008, which is when the deadline for submission is. By the way, the problem (in the senior section) is from the issue of Mathematical Reflections, 2007.

Problem: Find the least odd positive integer such that for each prime is divisible by at least four (distinct) primes.

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.