410241: GYM103990 E Etched Emerald Orbs

An archaeologist team found a tomb of the ancient tribe and discovered $$$2^{125}$$$ emerald orbs inside the tomb. The ancient tribe etched a numbers on each emerald orb. The archeologists spent two decades realizing that the ancient tribe etched each emerald orb with a unique number. Moreover, the numbers are from $$$1$$$ to $$$2^{125}$$$ in the ancient language.

Eddy, the only mathematician in the archaeologist team, recently figured out the relation between the number $$$k$$$ and the emerald orb numbered $$$k$$$. The weight of the emerald orb numbered $$$k$$$ is exactly $$$\frac{1}{k}$$$ grams. Since the number on each emerald orb is distinct from the number on any other emerald orb, there are no two emerald orbs having the same weight.

Eddy proposes a hypothesis: the ancient tribe used these emerald orbs to represent weight less than $$$1$$$ gram. It is trivial that the emerald orb numbered $$$k$$$ can represent $$$\frac{1}{k}$$$ gram. Then, Eddy tries to represent $$$\frac{2}{k}$$$ grams for $$$3\le k\le 4\times 10^{18}$$$ with two emerald orbs. He successfully finds that the emerald orbs numbered $$$2$$$ and $$$6$$$ can represent $$$\frac{2}{3}=\frac{1}{2}+\frac{1}{6}$$$ grams. Similarly, the emerald orbs numbered $$$3$$$ and $$$15$$$ can represent $$$\frac{2}{5}=\frac{1}{3}+\frac{1}{15}$$$ grams.

Can you write a program to help Eddy to check whether two emerald orbs can represent $$$\frac{2}{k}$$$ grams for a given integer $$$k$$$? If there are multiple combinations of two emerald orbs representing $$$\frac{2}{k}$$$ grams, output the combination minimizing the sum of the numbers etched on them. If there is no such combination, output -1.

The input contains only one positive integer $$$k$$$.

• $$$3\le k \le 4\times 10^{18}$$$.

If there is no solution, output -1. Otherwise, output two distinct integers $$$x$$$ and $$$y$$$ separated by a blank where $$$\frac{2}{k}=\frac{1}{x}+\frac{1}{y}$$$ and $$$1\le x<y\le 2^{125}$$$. If there are multiple solutions, output the solution minimizing $$$x+y$$$.

5

3 15

7

4 28