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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.