ZOJ3673:1729

1729 is the natural number following 1728 and preceding 1730. It is also known as the Hardy-Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding
a hospital visit to the Indian mathematician Srinivasa Ramanujan. In Hardy‘s words:

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting
number; it is the smallest number expressible as the sum of two (positive) cubes in two different ways."

The two different ways are these: 1729 = 1³ + 12³ = 9³ + 10³

Now your task is to count how many ways a positive number can be expressible as the sum of two positive cubes in. All the numbers in this task can be expressible as the sum of two positive cubes in at least one way.

Input

There‘re nearly 20,000 cases. Each case is a positive integer in a single line. And all these numbers are greater than 1 and less than 2⁶⁴.

Output

Please refer to the sample output. For each case, you should output a line. First the number of ways n. Then followed by n pairs of integer, (a_i,b_i),
indicating a way the given number can be expressible as the sum of a_i‘s cube and b_i‘s. (a_i≤ b_i, and a₁< a₂<
...< a_n)

Sample Input

9
4104
2622104000
21131226514944
48988659276962496

Sample Output

1 (1,2)
2 (2,16) (9,15)
3 (600,1340) (678,1322) (1020,1160)
4 (1539,27645) (8664,27360) (11772,26916) (17176,25232)
5 (38787,365757) (107839,362753) (205292,342952) (221424,336588) (231518,331954)

Hint

Although most numbers cannot be expressible as the sum of two positive cubes, the vast majority of numbers in this task can be expressible as the sum of two positive cubes in two or more ways.

题意比较简单，就不解释了

这题还真是弄了好久，再加上犯的各种小错误，但是终于算是搞出来了，不容易啊。。。

思路：要求满足的m=a³+b³=(a+b)(a²-ab+b²)的(a,b)组合。

令t=a+b,则t一定是m的约数，所以应枚举m的所有约数。

然后可以得到

a+b=t

ab=(t²-m/t)/3=p

继而转化为a²-ta+p=0是否有正整数解就可以了。

再就是注意范围要用unsigned long long。

#include <stdio.h>
#include <string.h>
#include <math.h>
#include <algorithm>
using namespace std;
#define ll unsigned long long
#define maxn 2642246//2^64次方的开立方
#define L 5000001
ll n;
ll prime[L],divi[500],e[500];
int len,len_prime,len_s;
bool flag[L] = {false};
struct node
{
    ll x,y;
} s[1005];

int cmp(node x,node y)
{
    return x.x<y.x;
}

void init()//素数且不说，所有合数都能分解成任意素数之积
{
    int i,j;
    len_prime = 0;
    for(i = 2; i<L; i++)
    {
        if(!flag[i]) prime[len_prime++] = i;//打下素数表
        for(int j = 0; j<len_prime && i*prime[j]<L; j++)
        {
            flag[i*prime[j]] = true;//所有非素数排除
            if(i%prime[j]==0) break;
        }
    }
}

void find_div(ll m)//找出m的所有素因子
{
    int i;
    ll k;
    len = 0;
    for(i = 0; i<len_prime && prime[i]*prime[i]<=m; i++)
    {
        if(m%prime[i] == 0)
        {
            divi[len] = prime[i];//因子保存
            k = 1;
            m/=prime[i];
            while(m%prime[i] == 0)//除尽为止
            {
                k++;//记录个数
                m/=prime[i];
            }
            e[len++] = k;
        }
    }
    if(m>1)
    {
        divi[len] = m;
        e[len++] = 1;
    }
}

ll can_sqrt(ll c)//要求整数解，b^2-4*a*c必须能开出整数
{
    ll r = sqrt(1.0*c);
    if(r*r == c) return r;
    return L;
}

int judge(ll x,ll y)//看这组解是否已经存在
{
    for(int i=0; i<len_s; i++)
        if(s[i].x==x&&s[i].y==y)
            return 1;
    return 0;
}

void solve(ll t)//找出解
{
    ll x1,x2;
    ll k = n/t;
    ll r = t*t-k;
    if(r>0 && r%3!=0) return ;
    r/=3;
    ll dis = t*t-4*r;
    if(dis<0) return ;
    ll c = can_sqrt(dis);
    if(c == L) return;
    if((t+c)%2 == 0)
    {
        x1 = (t+c)/2;
        x2 = t-x1;
        if(x1>x2) swap(x1,x2);
        if(x1>0&&x1<t&&x1<maxn&&x2<maxn&&x2>0&&x2<t&&!judge(x1,x2))
        {
            s[len_s].x=x1;
            s[len_s++].y=x2;
        }
    }
    if((t-c)>0 && (t-c)%2 == 0)
    {
        x1 = (t-c)/2;
        x2 = t-x1;
        if(x1>x2) swap(x1,x2);
        if(x1>0&&x1<t&&x1<maxn&&x2<maxn&&x2>0&&x2<t&&!judge(x1,x2))
        {
            s[len_s].x=x1;
            s[len_s++].y=x2;
        }
    }
}

ll ppow(ll a,ll b)
{
    ll ans = 1;
    while(b)
    {
        if(b&1) ans*=a;
        b>>=1;
        a*=a;
    }
    return ans;
}

void dfs(int m,ll sum)
{
    solve(sum);
    if(m>=len) return ;
    for(int i = 0; i<=e[m]; i++)//由个数去枚举次方，1为a,2为a^2,3为a^3，如此类推，枚举所有t
        dfs(m+1,sum*ppow(divi[m],i));
}

int main()
{
    init();
    while(~scanf("%llu",&n))
    {
        find_div(n);
        len_s = 0;
        dfs(0,1);
        sort(s,s+len_s,cmp);
        printf("%d",len_s);
        for(int i = 0; i<len_s; i++)
            printf(" (%llu,%llu)",s[i].x,s[i].y);
        printf("\n");
    }

    return 0;
}

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