1998 JBMO Problems/Problem 3
Find all pairs of positive integers such that
Solution
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.
See also
1998 JBMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 | ||
All JBMO Problems and Solutions |