1998 JBMO Problems/Problem 3
Revision as of 01:34, 18 September 2020 by Duck master (talk | contribs) (fixed stupid arithmetic mistake. the second solution to the eq is ok now)
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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