IEEEXtreme 10.0 - Goldbach's Second Conjecture

这是 meelo 原创的 IEEEXtreme极限编程大赛题解

Xtreme 10.0 - Goldbach's Second Conjecture

题目来源 第10届IEEE极限编程大赛

https://www.hackerrank.com/contests/ieeextreme-challenges/challenges/goldbachs-second-conjecture

An integer p > 1 is called a prime if its only divisors are 1 and p itself. A famous conjecture about primes is Goldbach's conjecture, which states that

Every even integer greater than 2 can be expressed as the sum of two primes.

The conjecture dates back to the year 1742, but still no one has been able to come up with a proof or find a counterexample to it. We considered asking you prove it here, but realized it would be too easy. Instead we present here a more difficult conjecture, known as Goldbach's second conjecture:

Every odd integer greater than 5 can be expressed as the sum of three primes.

In this problem we will provide you with an odd integer N greater than 5, and ask you to either find three primes p1, p2, p3 such that p1 + p2 + p3 = N, or inform us that N is a counterexample to Goldbach's second conjecture.

Input Format

The input contains a single odd integer 5 < N ≤ 10¹⁸.

Output Format

Output three primes, separated by a single space on a single line, whose sum is N. If there are multiple possible answers, output any one of them. If there are no possible answers, output a single line containing the text "counterexample" (without quotes).

Sample Input

65

Sample Output

23 31 11

Explanation

In the sample input N is 65. Consider the three integers 11, 23, 31. They are all prime, and their sum is 65. Hence they form a valid answer. That is, a line containing "11 23 31", "23 31 11", or any permutation of the three integers will be accepted. Other possible answers include "11 37 17" and "11 11 43".

题目解析

将一个奇数分解为三个质数，奇数最大有10¹⁸。可以遍历前两个质数，然后判断奇数与两个质数的差是否仍未质数。如果3个质数都有10¹⁷，那么肯定会超时。

事实上是，存在解前两个质数都不超过1000。这个时候关键的问题成为了，如何判断一个规模有10¹⁸的数为质数。常规的方法复杂度为O(sqrt(n))，会超时。这时候需要一点数论的知识，Miller–Rabin质数测试能够在O((logn)²)判断一个数是否为质数。算法在维基百科详细的介绍。下面程序里的Miller–Rabin质数测试使用的是github上的代码。

程序

C++

#include <cmath>
#include <cstdio>
#include <vector>
#include <iostream>
#include <algorithm>
#include <bitset>
using namespace std;

#define MAXN 1000
typedef unsigned long long ULL;
typedef long long LL;

bitset<MAXN> nums;
int primes[MAXN];
int num_prime = ;

void getPrimes(long long max) { // get all primes under max
    for(int i=; i<=sqrt(max+0.5); i++) {
        if(nums[i] == false) {
            primes[num_prime] = i;
            num_prime++;
            for(long long n=*i; n<max; n+=i) {
                nums[n] = true;
            }
        }
    }
    for(int i=int(sqrt(max+0.5))+; i<max; i++) {
        if(nums[i] == false) {
            primes[num_prime] = i;
            num_prime++;
        }
    }
}

LL MultiplyMod(LL a, LL b, LL mod) { //computes a * b % mod
    ULL r = ;
    a %= mod, b %= mod;
    while (b) {
        if (b & ) r = (r + a) % mod;
        b >>= , a = ((ULL) a << ) % mod;
    }
    return r;
}
template<typename T>
T PowerMod(T a, T n, T mod) {  //computes a^n % mod
    T r = ;
    while (n) {
        if (n & ) r = MultiplyMod(r, a, mod);
        n >>= , a = MultiplyMod(a, a, mod);
    }
    return r;
}
template<typename T>
bool isPrime(T n) { 
//determines if n is a prime number using Miller–Rabin primality test 
// from https://github.com/niklasb/tcr/blob/master/zahlentheorie/NumberTheory.cpp
    const int pn = , p[] = { , , , , , , , ,  };
    for (int i = ; i < pn; ++i)
        if (n % p[i] == ) return n == p[i];
    if (n < p[pn - ]) return ;
    T s = , t = n - ;
    while (~t & )
        t >>= , ++s;
    for (int i = ; i < pn; ++i) {
        T pt = PowerMod<T> (p[i], t, n);
        if (pt == ) continue;
        bool ok = ;
        for (int j = ; j < s && !ok; ++j) {
            if (pt == n - ) ok = ;
            pt = MultiplyMod(pt, pt, n);
        }
        if (!ok) return ;
    }
    return ;
}

int main() {
    long long n;
    cin >> n;

    getPrimes(MAXN);

    for(int i=; i<num_prime; i++) {
        for(int j=i; j<num_prime; j++) {
            if(isPrime(n-primes[j]-primes[i])) {
                printf("%lld %lld %lld", primes[i], primes[j], n-primes[i]-primes[j]);
                return ;
            }

        }
    }

    return ;
}

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