1998 JBMO Problems/Problem 3
Find all pairs of positive integers such that
Note that is at least one. Then is at least one, so .
Write , where . (We know that is nonnegative because .) Then our equation becomes . Taking logarithms base and dividing through by , we obtain .
Since divides the RHS of this equation, it must divide the LHS. Since by assumption, we must have , so that the equation reduces to , or . This equation has only the solutions and .
Therefore, our only solutions are , and , and we are done.
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