Power Pair

AtCoder

IDabc212_g

Time2000ms

Memory256MB

Difficulty—

Given is a prime number $P$. How many pairs of integers $(x, y)$ satisfy the following conditions? * $0 \leq x \leq P-1$ * $0 \leq y \leq P-1$ * There exists a positive integer $n$ such that $x^n \equiv y \pmod{P}$. Since the answer may be enormous, print it modulo $998244353$. ## Constraints * $2 \leq P \leq 10^{12}$ * $P$ is a prime number. ## Input Input is given from Standard Input in the following format: $P$ [samples]

Samples

Input #1

Output #1

4

Four pairs $(x, y) = (0, 0), (1, 1), (2, 1), (2, 2)$ satisfy the conditions.

Input #2

Output #2

Input #3

998244353

Output #3

329133417

API Response (JSON)

{
  "problem": {
    "name": "Power Pair",
    "description": {
      "content": "Given is a prime number $P$. How many pairs of integers $(x, y)$ satisfy the following conditions? *   $0 \\leq x \\leq P-1$ *   $0 \\leq y \\leq P-1$ *   There exists a positive integer $n$ such that $x",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc212_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given is a prime number $P$.\nHow many pairs of integers $(x, y)$ satisfy the following conditions?\n\n*   $0 \\leq x \\leq P-1$\n*   $0 \\leq y \\leq P-1$\n*   There exists a positive integer $n$ such that $x...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments