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前言
1.数论函数类Euclid
2.Euclid类的使用例子
3.资源
本博客所有文章分类的总目录：【总目录】本博客博文总目录-实时更新
开源Math.NET基础数学类库使用总目录：【目录】开源Math.NET基础数学类库使用总目录
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前言
数论就是指研究整数性质的一门理论。数论=算术。不过通常算术指数的计算，数论指数的理论。整数的基本元素是素数，所以数论的本质是对素数性质的研究。它是与平面几何同样历史悠久的学科。它大致包括代数数论、解析数论、计算数论等等。
Math.NET也包括了很多数论相关的函数，这些函数都是静态的，可以直接调用，如判断是否奇数，判断幂，平方数，最大公约数等等。同时部分函数已经作为扩展方法，可以直接在对象中使用。
如果本文资源或者显示有问题，请参考 本文原文地址：http://www.cnblogs.com/asxinyu/p/4301097.html
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1.数论函数类Euclid
Math.NET包括的数论函数除了一部分大家日常接触到的，奇数，偶数，幂，平方数，最大公约数，最小公倍数等函数，还有一些欧几里得几何的函数，当然也是比较简单的，这些问题的算法有一些比较简单，如最大公约数，最小公倍数等，都有成熟的算法可以使用，对于使用Math.NET的人来说，不必要了解太小，当然对于需要搞清楚原理的人来说，学习Math.NET的架构或者实现，是可以参考的。所以这里先给出Math.NET关于数论函数类Euclid的源代码：
1 ///2 /// 整数数论函数 3 /// Integer number theory functions. 4 /// 5 public static class Euclid 6 { 7 ///8 /// Canonical Modulus. The result has the sign of the divisor. 9 /// 10 public static double Modulus(double dividend, double divisor) 11 { 12 return ((dividend%divisor) + divisor)%divisor; 13 } 14 15 ///16 /// Canonical Modulus. The result has the sign of the divisor. 17 /// 18 public static float Modulus(float dividend, float divisor) 19 { 20 return ((dividend%divisor) + divisor)%divisor; 21 } 22 23 ///24 /// Canonical Modulus. The result has the sign of the divisor. 25 /// 26 public static int Modulus(int dividend, int divisor) 27 { 28 return ((dividend%divisor) + divisor)%divisor; 29 } 30 31 ///32 /// Canonical Modulus. The result has the sign of the divisor. 33 /// 34 public static long Modulus(long dividend, long divisor) 35 { 36 return ((dividend%divisor) + divisor)%divisor; 37 } 38 39 #if !NOSYSNUMERICS 40 ///41 /// Canonical Modulus. The result has the sign of the divisor. 42 /// 43 public static BigInteger Modulus(BigInteger dividend, BigInteger divisor) 44 { 45 return ((dividend%divisor) + divisor)%divisor; 46 } 47 #endif 48 49 ///50 /// Remainder (% operator). The result has the sign of the dividend. 51 /// 52 public static double Remainder(double dividend, double divisor) 53 { 54 return dividend%divisor; 55 } 56 57 ///58 /// Remainder (% operator). The result has the sign of the dividend. 59 /// 60 public static float Remainder(float dividend, float divisor) 61 { 62 return dividend%divisor; 63 } 64 65 ///66 /// Remainder (% operator). The result has the sign of the dividend. 67 /// 68 public static int Remainder(int dividend, int divisor) 69 { 70 return dividend%divisor; 71 } 72 73 ///74 /// Remainder (% operator). The result has the sign of the dividend. 75 /// 76 public static long Remainder(long dividend, long divisor) 77 { 78 return dividend%divisor; 79 } 80 81 #if !NOSYSNUMERICS 82 ///83 /// Remainder (% operator). The result has the sign of the dividend. 84 /// 85 public static BigInteger Remainder(BigInteger dividend, BigInteger divisor) 86 { 87 return dividend%divisor; 88 } 89 #endif 90 91 ///92 /// Find out whether the provided 32 bit integer is an even number. 93 /// 94 /// The number to very whether it's even. 95 ///True if and only if it is an even number. 96 public static bool IsEven(this int number) 97 { 98 return (number & 0x1) == 0x0; 99 }100 101 ///102 /// Find out whether the provided 64 bit integer is an even number.103 /// 104 /// The number to very whether it's even.105 ///True if and only if it is an even number. 106 public static bool IsEven(this long number)107 {108 return (number & 0x1) == 0x0;109 }110 111 ///112 /// Find out whether the provided 32 bit integer is an odd number.113 /// 114 /// The number to very whether it's odd.115 ///True if and only if it is an odd number. 116 public static bool IsOdd(this int number)117 {118 return (number & 0x1) == 0x1;119 }120 121 ///122 /// Find out whether the provided 64 bit integer is an odd number.123 /// 124 /// The number to very whether it's odd.125 ///True if and only if it is an odd number. 126 public static bool IsOdd(this long number)127 {128 return (number & 0x1) == 0x1;129 }130 131 ///132 /// Find out whether the provided 32 bit integer is a perfect power of two.133 /// 134 /// The number to very whether it's a power of two.135 ///True if and only if it is a power of two. 136 public static bool IsPowerOfTwo(this int number)137 {138 return number > 0 && (number & (number - 1)) == 0x0;139 }140 141 ///142 /// Find out whether the provided 64 bit integer is a perfect power of two.143 /// 144 /// The number to very whether it's a power of two.145 ///True if and only if it is a power of two. 146 public static bool IsPowerOfTwo(this long number)147 {148 return number > 0 && (number & (number - 1)) == 0x0;149 }150 151 ///152 /// Find out whether the provided 32 bit integer is a perfect square, i.e. a square of an integer.153 /// 154 /// The number to very whether it's a perfect square.155 ///True if and only if it is a perfect square. 156 public static bool IsPerfectSquare(this int number)157 {158 if (number 9)165 {166 return false; // return immediately in 6 cases out of 16.167 }168 169 if (lastHexDigit == 0 || lastHexDigit == 1 || lastHexDigit == 4 || lastHexDigit == 9)170 {171 int t = (int)Math.Floor(Math.Sqrt(number) + 0.5);172 return (t * t) == number;173 }174 175 return false;176 }177 178 ///179 /// Find out whether the provided 64 bit integer is a perfect square, i.e. a square of an integer.180 /// 181 /// The number to very whether it's a perfect square.182 ///True if and only if it is a perfect square. 183 public static bool IsPerfectSquare(this long number)184 {185 if (number 9)192 {193 return false; // return immediately in 6 cases out of 16.194 }195 196 if (lastHexDigit == 0 || lastHexDigit == 1 || lastHexDigit == 4 || lastHexDigit == 9)197 {198 long t = (long)Math.Floor(Math.Sqrt(number) + 0.5);199 return (t * t) == number;200 }201 202 return false;203 }204 205 ///206 /// Raises 2 to the provided integer exponent (0 <= exponent < 31).207 /// 208 /// The exponent to raise 2 up to.209 ///2 ^ exponent. 210 ///211 public static int PowerOfTwo(this int exponent)212 {213 if (exponent = 31)214 {215 throw new ArgumentOutOfRangeException("exponent");216 }217 218 return 1 222 /// Raises 2 to the provided integer exponent (0 <= exponent < 63).223 /// 224 /// The exponent to raise 2 up to.225 /// 2 ^ exponent. 226 ///227 public static long PowerOfTwo(this long exponent)228 {229 if (exponent = 63)230 {231 throw new ArgumentOutOfRangeException("exponent");232 }233 234 return ((long)1) 238 /// Find the closest perfect power of two that is larger or equal to the provided239 /// 32 bit integer.240 /// 241 /// The number of which to find the closest upper power of two.242 /// A power of two. 243 ///244 public static int CeilingToPowerOfTwo(this int number)245 {246 if (number == Int32.MinValue)247 {248 return 0;249 }250 251 const int maxPowerOfTwo = 0x40000000;252 if (number > maxPowerOfTwo)253 {254 throw new ArgumentOutOfRangeException("number");255 }256 257 number--;258 number |= number >> 1;259 number |= number >> 2;260 number |= number >> 4;261 number |= number >> 8;262 number |= number >> 16;263 return number + 1;264 }265 266 /// 267 /// Find the closest perfect power of two that is larger or equal to the provided268 /// 64 bit integer.269 /// 270 /// The number of which to find the closest upper power of two.271 ///A power of two. 272 ///273 public static long CeilingToPowerOfTwo(this long number)274 {275 if (number == Int64.MinValue)276 {277 return 0;278 }279 280 const long maxPowerOfTwo = 0x4000000000000000;281 if (number > maxPowerOfTwo)282 {283 throw new ArgumentOutOfRangeException("number");284 }285 286 number--;287 number |= number >> 1;288 number |= number >> 2;289 number |= number >> 4;290 number |= number >> 8;291 number |= number >> 16;292 number |= number >> 32;293 return number + 1;294 }295 296 /// 297 /// Returns the greatest common divisor ( 299 /// First Integer: a.300 /// Second Integer: b.301 ///gcd ) of two integers using Euclid's algorithm.298 ///Greatest common divisor 302 public static long GreatestCommonDivisor(long a, long b)303 {304 while (b != 0)305 {306 var remainder = a%b;307 a = b;308 b = remainder;309 }310 311 return Math.Abs(a);312 }313 314 ///gcd (a,b)315 /// Returns the greatest common divisor ( 318 /// List of Integers.319 ///gcd ) of a set of integers using Euclid's316 /// algorithm.317 ///Greatest common divisor 320 public static long GreatestCommonDivisor(IListgcd (list of integers)integers)321 {322 if (null == integers)323 {324 throw new ArgumentNullException("integers");325 }326 327 if (integers.Count == 0)328 {329 return 0;330 }331 332 var gcd = Math.Abs(integers[0]);333 334 for (var i = 1; (i 1); i++)335 {336 gcd = GreatestCommonDivisor(gcd, integers[i]);337 }338 339 return gcd;340 }341 342 /// 343 /// Returns the greatest common divisor ( 345 /// List of Integers.346 ///gcd ) of a set of integers using Euclid's algorithm.344 ///Greatest common divisor 347 public static long GreatestCommonDivisor(params long[] integers)348 {349 return GreatestCommonDivisor((IListgcd (list of integers))integers);350 }351 352 /// 353 /// Computes the extended greatest common divisor, such that a*x + b*y = 355 /// First Integer: a.356 /// Second Integer: b.357 /// Resulting x, such that a*x + b*y =gcd (a,b).354 ///gcd (a,b).358 /// Resulting y, such that a*x + b*y =gcd (a,b)359 ///Greatest common divisor 360 ///gcd (a,b)361 /// 368 public static long ExtendedGreatestCommonDivisor(long a, long b, out long x, out long y)369 {370 long mp = 1, np = 0, m = 0, n = 1;371 372 while (b != 0)373 {374 long rem;375 #if PORTABLE376 rem = a % b;377 var quot = a / b;378 #else379 long quot = Math.DivRem(a, b, out rem);380 #endif381 a = b;382 b = rem;383 384 var tmp = m;385 m = mp - (quot*m);386 mp = tmp;387 388 tmp = n;389 n = np - (quot*n);390 np = tmp;391 }392 393 if (a >= 0)394 {395 x = mp;396 y = np;397 return a;398 }399 400 x = -mp;401 y = -np;402 return -a;403 }404 405 ///362 /// long x,y,d;363 /// d = Fn.GreatestCommonDivisor(45,18,out x, out y);364 /// -> d == 9 && x == 1 && y == -2365 ///366 /// Thegcd of 45 and 18 is 9: 18 = 2*9, 45 = 5*9. 9 = 1*45 -2*18, therefore x=1 and y=-2.367 ///406 /// Returns the least common multiple ( 408 /// First Integer: a.409 /// Second Integer: b.410 ///lcm ) of two integers using Euclid's algorithm.407 ///Least common multiple 411 public static long LeastCommonMultiple(long a, long b)412 {413 if ((a == 0) || (b == 0))414 {415 return 0;416 }417 418 return Math.Abs((a/GreatestCommonDivisor(a, b))*b);419 }420 421 ///lcm (a,b)422 /// Returns the least common multiple ( 424 /// List of Integers.425 ///lcm ) of a set of integers using Euclid's algorithm.423 ///Least common multiple 426 public static long LeastCommonMultiple(IListlcm (list of integers)integers)427 {428 if (null == integers)429 {430 throw new ArgumentNullException("integers");431 }432 433 if (integers.Count == 0)434 {435 return 1;436 }437 438 var lcm = Math.Abs(integers[0]);439 440 for (var i = 1; i 449 /// Returns the least common multiple ( lcm ) of a set of integers using Euclid's algorithm.450 /// 451 /// List of Integers.452 ///Least common multiple 453 public static long LeastCommonMultiple(params long[] integers)454 {455 return LeastCommonMultiple((IListlcm (list of integers))integers);456 }457 458 #if !NOSYSNUMERICS459 /// 460 /// Returns the greatest common divisor ( 462 /// First Integer: a.463 /// Second Integer: b.464 ///gcd ) of two big integers.461 ///Greatest common divisor 465 public static BigInteger GreatestCommonDivisor(BigInteger a, BigInteger b)466 {467 return BigInteger.GreatestCommonDivisor(a, b);468 }469 470 ///gcd (a,b)471 /// Returns the greatest common divisor ( 473 /// List of Integers.474 ///gcd ) of a set of big integers.472 ///Greatest common divisor 475 public static BigInteger GreatestCommonDivisor(IListgcd (list of integers)integers)476 {477 if (null == integers)478 {479 throw new ArgumentNullException("integers");480 }481 482 if (integers.Count == 0)483 {484 return 0;485 }486 487 var gcd = BigInteger.Abs(integers[0]);488 489 for (int i = 1; (i BigInteger.One); i++)490 {491 gcd = GreatestCommonDivisor(gcd, integers[i]);492 }493 494 return gcd;495 }496 497 /// 498 /// Returns the greatest common divisor ( 500 /// List of Integers.501 ///gcd ) of a set of big integers.499 ///Greatest common divisor 502 public static BigInteger GreatestCommonDivisor(params BigInteger[] integers)503 {504 return GreatestCommonDivisor((IListgcd (list of integers))integers);505 }506 507 /// 508 /// Computes the extended greatest common divisor, such that a*x + b*y = 510 /// First Integer: a.511 /// Second Integer: b.512 /// Resulting x, such that a*x + b*y =gcd (a,b).509 ///gcd (a,b).513 /// Resulting y, such that a*x + b*y =gcd (a,b)514 ///Greatest common divisor 515 ///gcd (a,b)516 /// 523 public static BigInteger ExtendedGreatestCommonDivisor(BigInteger a, BigInteger b, out BigInteger x, out BigInteger y)524 {525 BigInteger mp = BigInteger.One, np = BigInteger.Zero, m = BigInteger.Zero, n = BigInteger.One;526 527 while (!b.IsZero)528 {529 BigInteger rem;530 BigInteger quot = BigInteger.DivRem(a, b, out rem);531 a = b;532 b = rem;533 534 BigInteger tmp = m;535 m = mp - (quot*m);536 mp = tmp;537 538 tmp = n;539 n = np - (quot*n);540 np = tmp;541 }542 543 if (a >= BigInteger.Zero)544 {545 x = mp;546 y = np;547 return a;548 }549 550 x = -mp;551 y = -np;552 return -a;553 }554 555 ///517 /// long x,y,d;518 /// d = Fn.GreatestCommonDivisor(45,18,out x, out y);519 /// -> d == 9 && x == 1 && y == -2520 ///521 /// Thegcd of 45 and 18 is 9: 18 = 2*9, 45 = 5*9. 9 = 1*45 -2*18, therefore x=1 and y=-2.522 ///556 /// Returns the least common multiple ( 558 /// First Integer: a.559 /// Second Integer: b.560 ///lcm ) of two big integers.557 ///Least common multiple 561 public static BigInteger LeastCommonMultiple(BigInteger a, BigInteger b)562 {563 if (a.IsZero || b.IsZero)564 {565 return BigInteger.Zero;566 }567 568 return BigInteger.Abs((a/BigInteger.GreatestCommonDivisor(a, b))*b);569 }570 571 ///lcm (a,b)572 /// Returns the least common multiple ( 574 /// List of Integers.575 ///lcm ) of a set of big integers.573 ///Least common multiple 576 public static BigInteger LeastCommonMultiple(IListlcm (list of integers)integers)577 {578 if (null == integers)579 {580 throw new ArgumentNullException("integers");581 }582 583 if (integers.Count == 0)584 {585 return 1;586 }587 588 var lcm = BigInteger.Abs(integers[0]);589 590 for (int i = 1; i 599 /// Returns the least common multiple ( lcm ) of a set of big integers.600 /// 601 /// List of Integers.602 ///Least common multiple 603 public static BigInteger LeastCommonMultiple(params BigInteger[] integers)604 {605 return LeastCommonMultiple((IListlcm (list of integers))integers);606 }607 #endif608 }
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2.Euclid类的使用例子
上面已经看到源码，也提到了，Euclid作为静态类，其中的很多静态方法都可以直接作为扩展方法使用。这里看看几个简单的例子：
1 // 1. Find out whether the provided number is an even number 2 Console.WriteLine(@"1.判断提供的数字是否是偶数"); 3 Console.WriteLine(@"{0} 是偶数 = {1}. {2} 是偶数 = {3}", 1, Euclid.IsEven(1), 2, 2.IsEven()); 4 Console.WriteLine(); 5 6 // 2. Find out whether the provided number is an odd number 7 Console.WriteLine(@"2.判断提供的数字是否是奇数"); 8 Console.WriteLine(@"{0} 是奇数 = {1}. {2} 是奇数 = {3}", 1, 1.IsOdd(), 2, Euclid.IsOdd(2)); 9 Console.WriteLine();10 11 // 3. Find out whether the provided number is a perfect power of two12 Console.WriteLine(@"2.判断提供的数字是否是2的幂");13 Console.WriteLine(@"{0} 是2的幂 = {1}. {2} 是2的幂 = {3}", 5, 5.IsPowerOfTwo(), 16, Euclid.IsPowerOfTwo(16));14 Console.WriteLine();15 16 // 4. Find the closest perfect power of two that is larger or equal to 9717 Console.WriteLine(@"4.返回大于等于97的最小2的幂整数");18 Console.WriteLine(97.CeilingToPowerOfTwo());19 Console.WriteLine();20 21 // 5. Raise 2 to the 1622 Console.WriteLine(@"5. 2的16次幂");23 Console.WriteLine(16.PowerOfTwo());24 Console.WriteLine();25 26 // 6. Find out whether the number is a perfect square27 Console.WriteLine(@"6. 判断提供的数字是否是平方数");28 Console.WriteLine(@"{0} 是平方数 = {1}. {2} 是平方数 = {3}", 37, 37.IsPerfectSquare(), 81, Euclid.IsPerfectSquare(81));29 Console.WriteLine();30 31 // 7. Compute the greatest common divisor of 32 and 36 32 Console.WriteLine(@"7. 返回32和36的最大公约数");33 Console.WriteLine(Euclid.GreatestCommonDivisor(32, 36));34 Console.WriteLine();35 36 // 8. Compute the greatest common divisor of 492, -984, 123, 24637 Console.WriteLine(@"8. 返回的最大公约数：492、-984、123、246");38 Console.WriteLine(Euclid.GreatestCommonDivisor(492, -984, 123, 246));39 Console.WriteLine();40 41 // 9. Compute the least common multiple of 16 and 1242 Console.WriteLine(@"10. 计算的最小公倍数：16和12");43 Console.WriteLine(Euclid.LeastCommonMultiple(16, 12));44 Console.WriteLine();
结果如下：
1.判断提供的数字是否是偶数1 是偶数 = False. 2 是偶数 = True2.判断提供的数字是否是奇数1 是奇数 = True. 2 是奇数 = False2.判断提供的数字是否是2的幂5 是2的幂 = False. 16 是2的幂 = True4.返回大于等于97的最小2的幂整数1285. 2的16次幂655366. 判断提供的数字是否是平方数37 是平方数 = False. 81 是平方数 = True7. 返回32和36的最大公约数48. 返回的最大公约数：492、-984、123、24612310. 计算的最小公倍数：16和1248
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来自 “ ITPUB博客 ” ，链接：https://blog.itpub.net/97/viewspace-2806729/ ，如需转载，请注明出处，否则将追究法律责任。
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