Discrete Logarithm Problems

AtCoder

IDabc335_g

Time5000ms

Memory256MB

Difficulty—

You are given $N$ integers $A_1,\ldots,A_N$ and a prime number $P$. Find the number of pairs of integers $(i,j)$ that satisfy both of the following conditions: * $1 \leq i,j \leq N$; * There is a positive integer $k$ such that $A_i^k \equiv A_j \mod P$. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $1 \leq A_i < P$ * $2 \leq P \leq 10^{13}$ * $P$ is prime. * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $P$ $A_1$ $\ldots$ $A_N$ [samples]

Samples

Input #1

3 13
2 3 5

Output #1

5

Five pairs satisfy the conditions: $(1,1),(1,2),(1,3),(2,2),(3,3)$.
For example, for the pair $(1,3)$, if we take $k=9$, then $A_1^9 = 512 \equiv 5 = A_3 \mod 13$.

Input #2

5 2
1 1 1 1 1

Output #2

Input #3

10 9999999999971
141592653589 793238462643 383279502884 197169399375 105820974944 592307816406 286208998628 34825342117 67982148086 513282306647

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Discrete Logarithm Problems",
    "description": {
      "content": "You are given $N$ integers $A_1,\\ldots,A_N$ and a prime number $P$. Find the number of pairs of integers $(i,j)$ that satisfy both of the following conditions: *   $1 \\leq i,j \\leq N$; *   There is a",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 5000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc335_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given $N$ integers $A_1,\\ldots,A_N$ and a prime number $P$. Find the number of pairs of integers $(i,j)$ that satisfy both of the following conditions:\n\n*   $1 \\leq i,j \\leq N$;\n*   There is a...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments