A student I’m tutoring was working unsuccessfully on the following problem from the 2011 Mandelbrot Competition:
Let be positive integers such that . Find the minimum value of .
After some tinkering, I concluded that the problem as stated has no solution. I am now almost certain it was printed incorrectly: should be replaced by , and then we can solve the problem as follows:
It follows that . Since are positive integers we must have , and then it follows that the smallest solution occurs when . But what I’d like to discuss, briefly, is the argument showing that the misprinted problem has no solution.