1998 JBMO Problems/Problem 3

Find all pairs of positive integers $(x,y)$ such that $x^y = y^{x - y}.$

Solution

Note that $x^y$ is at least one. Then $y^{x - y}$ is at least one, so $x \geq y$ .

Write $x = a^{b+c}, y = a^c$ , where $\gcd(b, c) = 1$ . (We know that $b$ is nonnegative because $x\geq y$ .) Then our equation becomes $a^{(b+c)*a^c} = a^{c*(a^{b+c} - a^c)}$ . Taking logarithms base $a$ and dividing through by $a^c$ , we obtain $b + c = c*(a^b - 1)$ .

Since $c$ divides the RHS of this equation, it must divide the LHS. Since $\gcd(b, c) = 1$ by assumption, we must have $c = 1$ , so that the equation reduces to $b + 1 = a^b - 1$ , or $b + 2 = a^b$ . This equation has only the solutions $b = 1, a = 3$ and $b = 2, a = 2$ .

Therefore, our only solutions are $x = 3^{1 + 1} = 9, y = 3^1 = 3$ , and $x = 2^{2+1} = 8, y = 2^2 = 4$ , and we are done.

1998 JBMO (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5
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1998 JBMO Problems/Problem 3

Solution

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