Equation

AtCoder

IDarc158_d

Time4000ms

Memory256MB

Difficulty—

You are given a positive integer $n$, and a prime number $p$ at least $5$. Find a triple of integers $(x,y,z)$ that satisfies all of the following conditions. * $1\leq x < y < z \leq p - 1$. * $(x+y+z)(x^n+y^n+z^n)(x^{2n}+y^{2n}+z^{2n}) \equiv x^{3n}+y^{3n}+z^{3n}\pmod{p}$. It can be proved that such a triple $(x,y,z)$ always exists. You have $T$ test cases to solve. ## Constraints * $1\leq T\leq 10^5$ * $1\leq n\leq 10^9$ * $p$ is a prime number satisfying $5\leq p\leq 10^9$. ## Input The input is given from Standard Input in the following format: $T$ $\text{case}_1$ $\vdots$ $\text{case}_T$ Each case is in the following format: $n$ $p$ [samples]

Samples

Input #1

3
1 7
2 7
10 998244353

Output #1

1 4 6
1 2 5
20380119 21549656 279594297

For the first test case:

*   $(x+y+z)(x^n+y^n+z^n)(x^{2n}+y^{2n}+z^{2n}) = (1+4+6)(1+4+6)(1+16+36) = 6413$, and
*   $x^{3n}+y^{3n}+z^{3n} = 1 + 64 + 216 = 281$.

We have $6413\equiv 281\pmod{7}$, so the conditions are satisfied.

API Response (JSON)

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