題目描述
已知多項式方程:\(a_0 + a_1x + a_2x^2+...+a_nx^n = 0\)
求這個方程在[1,m]內的整數解
\(1\leq n\leq100,|a_i|\leq 10^{10000},a_n≠0,m\leq 10^6\)
Solution
首先由於數據過大,只能字符串讀入了hhh。然後:對於每個x,算出f(x)%pi,如果f(x)=0則f(x)%pi必然=0,多選幾個素數,就可以在一定範圍大小內判斷成功。好像不夠快?
\(f(x+p)≡f(x)(mod p)\)
對於一個x,\(f(x)≠0(mod p)\),則x+p,x+2p……均不爲方程的解
有了這個性質就可以瞎搞了。取幾個較大的質數搞一搞……
Code
#include <iostream>
#include <cstdio>
using namespace std;
inline long long read() {
long long x = 0; int f = 0; char c = getchar();
while (c < '0' || c > '9') f |= c == '-', c = getchar();
while (c >= '0' && c <= '9') x = (x << 1) + (x << 3) + (c ^ 48), c = getchar();
return f? -x:x;
}
long long a[110][4], mod[4] = {10007, 11003, 12007, 13001};
int b[1000010], n, m;
string qwq;
bool check(long long x) {
for (int k = 0; k < 4; k++) {
long long tmp = a[n][k];
for (int i = n - 1; i >= 0; i--)
tmp = (tmp * x + a[i][k]) % mod[k];
if (tmp) {
for (int i = x; i <= m; i += mod[k]) b[i] = 1;
return 0;
}
}
return 1;
}
int main() {
n = read(); m = read();
for (int i = 0; i <= n; ++i) {
cin >> qwq;
for (int k = 0; k < 4; ++k) {
long long flg = 1, tmp = 0;
for (int j = 0; j < qwq.length(); ++j)
if (qwq[j] == '-') flg = -1;
else tmp = ((tmp << 1) + (tmp << 3) + (qwq[j] ^ 48)) % mod[k];
a[i][k] = tmp * flg;
}
}
int ans = 0;
for (int x = 1; x <= m; x++)
if (!b[x] && check(x)) ans++;
printf("%d\n", ans);
for (int i = 1; i <= m; i++)
if (!b[i]) printf("%d\n", i);
return 0;
}