Tokitsukaze and Powers

题意翻译

给出三个整数 $n,m,p$，试求出 $n$ 个互不相同的整数使得它们中的每个整数 $x$ 均满足以下条件： - $x \in [0,m-1]$ - $\gcd(m,x) = 1$ - $\nexists e,k \in \mathbb{N},p^e = km+x$ 如果能够找到，输出可能的选择方案，否则输出 `-1`。 $1 \leq n \leq 5 \times 10^5 , 1 \leq p < m \leq 10^{18}$，保证 $m$ 是某个质数的正整数次幂。

题目描述

Tokitsukaze is playing a room escape game designed by SkywalkerT. In this game, she needs to find out hidden clues in the room to reveal a way to escape. After a while, she realizes that the only way to run away is to open the digital door lock since she accidentally went into a secret compartment and found some clues, which can be interpreted as: - Only when you enter $ n $ possible different passwords can you open the door; - Passwords must be integers ranged from $ 0 $ to $ (m - 1) $ ; - A password cannot be $ x $ ( $ 0 \leq x < m $ ) if $ x $ and $ m $ are not coprime (i.e. $ x $ and $ m $ have some common divisor greater than $ 1 $ ); - A password cannot be $ x $ ( $ 0 \leq x < m $ ) if there exist non-negative integers $ e $ and $ k $ such that $ p^e = k m + x $ , where $ p $ is a secret integer; - Any integer that doesn't break the above rules can be a password; - Several integers are hidden in the room, but only one of them can be $ p $ . Fortunately, she finds that $ n $ and $ m $ are recorded in the lock. However, what makes Tokitsukaze frustrated is that she doesn't do well in math. Now that she has found an integer that is suspected to be $ p $ , she wants you to help her find out $ n $ possible passwords, or determine the integer cannot be $ p $ .

输入输出格式

输入格式

The only line contains three integers $ n $ , $ m $ and $ p $ ( $ 1 \leq n \leq 5 \times 10^5 $ , $ 1 \leq p < m \leq 10^{18} $ ). It is guaranteed that $ m $ is a positive integer power of a single prime number.

输出格式

If the number of possible different passwords is less than $ n $ , print a single integer $ -1 $ . Otherwise, print $ n $ distinct integers ranged from $ 0 $ to $ (m - 1) $ as passwords. You can print these integers in any order. Besides, if there are multiple solutions, print any.

输入输出样例

输入样例 #1

1 2 1

输出样例 #1

-1

输入样例 #2

3 5 1

输出样例 #2

2 4 3

输入样例 #3

2 5 4

输出样例 #3

2 3

输入样例 #4

4 9 8

输出样例 #4

2 4 7 5

说明

In the first example, there is no possible password. In each of the last three examples, the given integer $ n $ equals to the number of possible different passwords for the given integers $ m $ and $ p $ , so if the order of numbers in the output is ignored, the solution is unique as shown above.