[NBUT 1224 Happiness Hotel 佩尔方程最小正整数解]连分数法解Pell方程
题意：求方程x2-Dy2=1的最小正整数解
思路：用连分数法解佩尔方程，关键是找出√d的连分数表示的循环节。具体过程参见：http://m.blog.csdn.net/blog/wh2124335/8871535
- 当d为完全平方数时无解
- 将√d表示成连分数的形式，例如：
- 当d不为完全平方数时，√d为无理数，那么√d总可以表示成：
- 记
- 当n为偶数时，x0=p，y0=q；当n为奇数时，x0=2p2+1，y0=2pq
求d在1000以内佩尔方程的最小正整数解的c++打表程序（正常跑比较慢，这个题需要离线打表）：
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 |
#pragma comment(linker, "/STACK:10240000") #include <map> #include <set> #include <cmath> #include <ctime> #include <deque> #include <queue> #include <stack> #include <vector> #include <cstdio> #include <string> #include <cstdlib> #include <cstring> #include <iostream> #include <algorithm> using namespace std; #define X first #define Y second #define pb push_back #define mp make_pair #define all(a) (a).begin(), (a).end() #define fillchar(a, x) memset(a, x, sizeof(a)) #define copy(a, b) memcpy(a, b, sizeof(a)) typedef long long ll; typedef pair<int, int> pii; typedef unsigned long long ull; #ifndef ONLINE_JUDGE void RI(vector<int>&a,int n){a.resize(n);for(int i=0;i<n;i++)scanf("%d",&a[i]);} void RI(){}void RI(int&X){scanf("%d",&X);}template<typename...R> void RI(int&f,R&...r){RI(f);RI(r...);}void RI(int*p,int*q){int d=p<q?1:-1; while(p!=q){scanf("%d",p);p+=d;}}void print(){cout<<endl;}template<typename T> void print(const T t){cout<<t<<endl;}template<typename F,typename...R> void print(const F f,const R...r){cout<<f<<", ";print(r...);}template<typename T> void print(T*p, T*q){int d=p<q?1:-1;while(p!=q){cout<<*p<<", ";p+=d;}cout<<endl;} #endif template<typename T>bool umax(T&a, const T&b){return b<=a?false:(a=b,true);} template<typename T>bool umin(T&a, const T&b){return b>=a?false:(a=b,true);} const double PI = acos(-1.0); const int INF = 1e9 + 7; const double EPS = 1e-12; /* -------------------------------------------------------------------------------- */ struct BigInt { const static int maxI = 1e8; const static int Len = 8; typedef vector<int> vi; typedef long long LL; vi num; bool symbol; BigInt() { num.clear(); symbol = 0; } BigInt(int x) { symbol = 0; if (x < 0) { symbol = 1; x = -x; } num.push_back(x % maxI); if (x >= maxI) num.push_back(x / maxI); } BigInt(bool s, vi x) { symbol = s; num = x; } BigInt(char s[]) { int len = strlen(s), x = 1, sum = 0, p = s[0] == '-'; symbol = p; for (int i = len - 1; i >= p; i--) { sum += (s[i] - '0') * x; x *= 10; if (x == 1e8 || i == p) { num.push_back(sum); sum = 0; x = 1; } } while (num.back() == 0 && num.size() > 1) num.pop_back(); } void push(int x) { num.push_back(x); } BigInt abs() const { return BigInt(false, num); } bool smaller(const vi &a, const vi &b) const { if (a.size() != b.size()) return a.size() < b.size(); for (int i = a.size() - 1; i >= 0; i--) { if (a[i] != b[i]) return a[i] < b[i]; } return 0; } bool operator < (const BigInt &p) const { if (symbol && !p.symbol) return true; if (!symbol && p.symbol) return false; if (symbol && p.symbol) return smaller(p.num, num); return smaller(num, p.num); } bool operator > (const BigInt &p) const { return p < *this; } bool operator == (const BigInt &p) const { return !(p < *this) && !(*this < p); } bool operator != (const BigInt &p) const { return *this < p || p < *this; } bool operator >= (const BigInt &p) const { return !(*this < p); } bool operator <= (const BigInt &p) const { return !(p < *this); } vi add(const vi &a, const vi &b) const { vi c; c.clear(); int x = 0; for (int i = 0; i < a.size(); i++) { x += a[i]; if (i < b.size()) x += b[i]; c.push_back(x % maxI); x /= maxI; } for (int i = a.size(); i < b.size(); i++) { x += b[i]; c.push_back(x % maxI); x /= maxI; } if (x) c.push_back(x); while (c.back() == 0 && c.size() > 1) c.pop_back(); return c; } vi sub(const vi &a, const vi &b) const { vi c; c.clear(); int x = 1; for (int i = 0; i < b.size(); i++) { x += maxI + a[i] - b[i] - 1; c.push_back(x % maxI); x /= maxI; } for (int i = b.size(); i < a.size(); i++) { x += maxI + a[i] - 1; c.push_back(x % maxI); x /= maxI; } while (c.back() == 0 && c.size() > 1) c.pop_back(); return c; } vi mul(const vi &a, const vi &b) const { vi c; c.resize(a.size() + b.size()); for (int i = 0; i < a.size(); i++) { for (int j = 0; j < b.size(); j++) { LL tmp = (LL)a[i] * b[j] + c[i + j]; c[i + j + 1] += tmp / maxI; c[i + j] = tmp % maxI; } } while (c.back() == 0 && c.size() > 1) c.pop_back(); return c; } vi div(const vi &a, const vi &b) const { vi c(a.size()), x(1, 0), y(1, 0), z(1, 0), t(1, 0); y.push_back(1); for (int i = a.size() - 1; i >= 0; i--) { z[0] = a[i]; x = add(mul(x, y), z); if (smaller(x, b)) continue; int l = 1, r = maxI - 1; while (l < r) { int m = (l + r + 1) >> 1; t[0] = m; if (smaller(x, mul(b, t))) r = m - 1; else l = m; } c[i] = l; t[0] = l; x = sub(x, mul(b, t)); } while (c.back() == 0 && c.size() > 1) c.pop_back(); return c; } BigInt operator + (const BigInt &p) const { if (!symbol && !p.symbol) return BigInt(false, add(num, p.num)); if (!symbol && p.symbol) { return *this >= p.abs() ? BigInt(false, sub(num, p.num)) : BigInt(true, sub(p.num, num)); } if (symbol && !p.symbol) { return (*this).abs() > p ? BigInt(true, sub(num, p.num)) : BigInt(false, sub(p.num, num)); } return BigInt(true, add(num, p.num)); } BigInt operator - (const BigInt &p) const { return *this + BigInt(!p.symbol, p.num); } BigInt operator * (const BigInt &p) const { BigInt res(symbol ^ p.symbol, mul(num, p.num)); if (res.symbol && res.num.size() == 1 && res.num[0] == 0) res.symbol = false; return res; } BigInt operator / (const BigInt &p) const { if (p == BigInt(0)) return p; BigInt res(symbol ^ p.symbol, div(num, p.num)); if (res.symbol && res.num.size() == 1 && res.num[0] == 0) res.symbol = false; return res; } BigInt operator % (const BigInt &p) const { return *this - *this / p * p; } void show() const { if (symbol) putchar('-'); printf("%d", num[num.size() - 1]); for (int i = num.size() - 2; i >= 0; i--) { printf("%08d", num[i]); } //putchar('\n'); } int TotalDigit() const { int x = num[num.size() - 1] / 10, t = 1; while (x) { x /= 10; t++; } return t + (num.size() - 1) * Len; } }; template<typename T> T gcd(T a, T b) { return b == 0? a : gcd(b, a % b); } template<typename T> struct Fraction { T a, b; Fraction(T a, T b) { T g = gcd(a, b); this->a = a / g; this->b = b / g; if (this->b < 0) { this->a = this->a * T(- 1); this->b = this->b * T(- 1); } } Fraction(T a) { this->a = a; this->b = 1; } Fraction() {} Fraction operator + (const Fraction &that) const { T x = a * that.b + b * that.a, y = b * that.b; return Fraction(x, y); } Fraction operator - (const Fraction &that) const { T x = a * that.b - b * that.a, y = b * that.b; return Fraction(x, y); } Fraction operator * (const Fraction &that) const { T x = a * that.a, y = b * that.b; return Fraction(x, y); } Fraction operator / (const Fraction &that) const { T x = a * that.b, y = b * that.a; return Fraction(x, y); } Fraction operator += (const Fraction &that) { return *this = *this + that; } Fraction operator -= (const Fraction &that) { return *this = *this - that; } Fraction operator *= (const Fraction &that) { return *this = *this * that; } Fraction operator /= (const Fraction &that) { return *this = *this / that; } Fraction operator ! () const { return Fraction(b, a); } bool operator == (const Fraction &that) const { return a == that.a && b == that.b; } bool operator != (const Fraction &that) const { return a != that.a || b != that.b; } }; template<typename T> T getInt(Fraction<T> a, T d, Fraction<T> b) { T Min = 0, Max; Fraction<T> buf = a * d + b; Max = buf.a / buf.b; while (Min < Max) { T Mid = (Min + Max + 1) / 2; buf = (b - Mid) * (b - Mid); buf = buf / a / a; if (buf.a <= buf.b * d) Min = Mid; else Max = Mid - 1; } return Min; } void work(int n) { int k = (int)sqrt(n + 0.5); if (k * k == n) { printf("no solution"); return ; } Fraction<BigInt> a(1), b(0), aa, bb; BigInt d(n); vector<BigInt> R; BigInt t = getInt(a, d, b); aa = a / (a * a * d - (b - t) * (b - t)); bb = (b - t) * BigInt(- 1) / (a * a * d - (b - t) * (b - t)); a = aa; b = bb; do { R.pb(t); t = getInt(a, d, b); aa = a / (a * a * d - (b - t) * (b - t)); bb = (b - t) * BigInt(- 1) / (a * a * d - (b - t) * (b - t)); a = aa; b = bb; } while (t != R[0] * 2); Fraction<BigInt> ans(R[R.size() - 1]); for (int i = 1; i < R.size(); i ++) { ans = !ans + R[R.size() - i - 1]; } BigInt x0 = ans.a, y0 = ans.b; if (R.size() & 1) { x0 = ans.a * ans.a * 2 + 1; y0 = ans.a * ans.b * 2; } x0.show(); } int main() { #ifndef ONLINE_JUDGE freopen("in.txt", "r", stdin); freopen("out.txt", "w", stdout); #endif // ONLINE_JUDGE int n; puts("char ans[][100] = {\"\", "); for (int i = 1; i <= 1000; i ++) { printf("\""); work(i); printf("\", "); if (i % 20 == 0) puts(""); } puts("\n};"); return 0; } |
