[从头学数学] 第205节 初等数论初步

剧情提要：
[机器小伟]在[工程师阿伟]的陪同下进入了[九转金丹]之第七转的修炼。
这次要研究的是[初等数论初步]。

正剧开始：

星历2016年05月08日 10:05:37, 银河系厄尔斯星球中华帝国江南行省。
[工程师阿伟]正在和[机器小伟]一起研究[初等数论初步]。

<span style="font-size:18px;">710316能被3整除710316能被9整除710316 不能 被11整除710316 不能 被7整除#根据规律判整除性#其实并不实用，只是说明一下规律def tmp():    number = 710316;    array =  numberArray(number);    size = len(array);    #被11整除    array_odd = array[0:size:2];    array_even = array[1:size:2];    #被7整除    array_last3 = number%1000;    array_front = (number-array_last3)//1000;    #判断被3整除    if sum(array)%3==0:        print('{0}能被3整除'.format(number));    else:        print('{0} 不能 被3整除'.format(number));    #判断被9整除    if sum(array)%9==0:        print('{0}能被9整除'.format(number));    else:        print('{0} 不能 被9整除'.format(number));    #判断被11整除    if (sum(array_odd)-sum(array_even))%11 == 0:        print('{0}能被11整除'.format(number));    else:        print('{0} 不能 被11整除'.format(number));            #判断被7整除    if (array_last3 - array_front)%7==0:        print('{0}能被7整除'.format(number));    else:        print('{0} 不能 被7整除'.format(number));#把一个整数的各位数字放入数组#由个位往高位排列def numberArray(number):    array = [];    bit = len(str(number));    for i in range(bit):        array.append(number%10);        number//=10;    #array.reverse();        return array;</span>

<span style="font-size:18px;">#判断一个数是否质数  def prime(num):      if (num < 2):          return False;            sqr = int(math.sqrt(num))+1;        for i in range(2, sqr):          if (num%i==0):              return False;        return True; </span>

<span style="font-size:18px;">#最大公约数def gcd(m, n):        m, n = max(m, n), min(m, n);        i = 1;        while n:            print('step', i, ': ', m, n);            i += 1;            m, n = n, m % n                    return m</span>

<span style="font-size:18px;">#最小公倍数def lcm(m, n):    return m*n/gcd(m, n);</span>

<span style="font-size:18px;">>>> 2625.0print(lcm(375, 105));</span>

<span style="font-size:18px;">#分解质因数  def primeFactor(num, lists):      if (num < 2):          lists.append(num);          return lists;      elif (prime(num) == True):          lists.append(num);          return lists;      else:              sqr = int(math.sqrt(num))+1;          i = 2;            while i <= sqr:              if (num % i == 0 and prime(i) == True):                  lists.append(i);                  num = num//i;                  break;                            i+=1;          return primeFactor(num, lists); </span>

<span style="font-size:18px;">>>> 720 = 2 * 2 * 2 * 2 * 3 * 3 * 5152 = 2 * 2 * 2 * 19813680.0def tmp2():    number = [720, 152];    for j in range(len(number)):        factors = primeFactor(number[j], []);        count = len(factors);        s = str(number[j])+' = ';                for i in range(count):            s += str(factors[i]);            if (i < count-1):                s += ' * ';        print(s);            print(gcd(720,152));    print(lcm(720, 152));</span>

<span style="font-size:18px;">>>> 21的欧拉函数值φ(21) = 12.0def tmp3(number):    print('{0}的欧拉函数值φ({0}) = {1}'.format(number, round(eulerPhi(number),3)))#欧拉函数def eulerPhi(number):    factors = set(primeFactor(number, []));    factors = sorted(factors);        #print(factors);    phi = number;    for i in range(len(factors)):        phi *= (1-1/factors[i]);    return phi;</span>

<span style="font-size:18px;">>>> 25的欧拉函数值φ(25) = 20.0def tmp4():    for i in range(1, 21):        if (18**i)%25==1:            print(i);            break;</span>

<span style="font-size:18px;">>>> [4.0, 9.0, 14.0]def tmp5():    result = [];    mod = 15;    remain = 6;    quotient = 9;            for i in range(1, 20):                x = (mod*i + remain)/quotient;        if abs((int(x))-x)< 1e-6:            result.append(x);        if x > mod:            break;    print(result);</span>

<span style="font-size:18px;">>>> [9.0]#例6def tmp5():    result = [];    mod = 74;    remain = 1;    quotient = 33;            for i in range(1, 20):                x = (mod*i + remain)/quotient;        if abs((int(x))-x)< 1e-6:            result.append(x);        if x > mod:            break;    print(result);</span>

<span style="font-size:18px;">var s = ['f(a)=e, f(b)=f, f(c)=g','a=-1,  e=2;','b=0,  f=3;','c=1,  g=6;','f(x) = 2(x-0)(x-1)/(-1)/(-2)',' +3(x+1)(x-1)/(1)/(-1)',' +6(x+1)(x-0)/(2)/(1)',];</span>

<span style="font-size:18px;">[58, 238, 418, 598, 778, 958, 1138, 1318, 1498]#孙子定理的变通def tmp6():    remain = [2, 3, 4];    mod = [4, 5, 9];    count = len(mod);    multi = 1;    for i in range(count):        multi *= mod[i];    #直接遍历这个范围    bound = sum(remain)*multi+1;    result = [];    for i in range(1, bound):        for j in range(count):            if i%mod[j]!=remain[j]:                break;            if j >= count-1:                result.append(i);        print(result);</span>

<span style="font-size:18px;">[282, 975, 1668, 2361, 3054, 3747, 4440, 5133, 5826, 6519, 7212, 7905]#孙子定理的变通def tmp6():    #模数    mod = [7, 9, 11];    #余数    remain = [2, 3, 7];        count = len(mod);    multi = 1;    for i in range(count):        multi *= mod[i];    #直接遍历这个范围    bound = sum(remain)*multi+1;    result = [];    for i in range(1, bound):        for j in range(count):            if i%mod[j]!=remain[j]:                break;            if j >= count-1:                result.append(i);        print(result);</span>

<span style="font-size:18px;">[2111, 4421, 6731, 9041, 11351, 13661, 15971, 18281, 20591, 22901, 25211, 27521, 29831, 32141, 34451, 36761, 39071, 41381, 43691, 46001]#韩信点兵#孙子定理的变通def tmp6():    #模数    mod = [5, 6, 7, 11];    #余数    remain = [1, 5, 4, 10];        count = len(mod);    multi = 1;    for i in range(count):        multi *= mod[i];    #直接遍历这个范围    bound = sum(remain)*multi+1;    result = [];    for i in range(1, bound):        for j in range(count):            if i%mod[j]!=remain[j]:                break;            if j >= count-1:                result.append(i);        print(result);</span>

<span style="font-size:18px;">2 30 685 25 708 20 7211 15 7414 10 7617 5 7820 0 80#百马百瓦def tmp8():    a = 33;    b = 50;    c = 200;    result = [];        for i in range(a+1):        for j in range(b+1):            for k in range(c+1):                if (abs(((i*3 + j*2 + k*0.5) - 100)) < 1e-6) and i+j+k == 100:                    print(i, j, k);</span>

<span style="font-size:18px;">#不定方程def tmp7():    #ax + by  = c    #b为大于1的整数    a, b, c = 3, 2, 5;    u, v, i, j = a, b, 0, 1;    r = u%v;    k = j;        while r!=1:        u, v = v, r;        q = u//v;        r = u%v;        j = i-q*j;        i = k;        k = j;    x = c*k;    y = c*(1-a*k)/b;    print(x, y);</span>

>>> -34 6.0#不定方程def tmp7():    #ax + by  = c    #b为大于1的整数    a, b, c = 13, 74, 2;    u, v, i, j = a, b, 0, 1;    r = u%v;    k = j;        while r!=1:        u, v = v, r;        q = u//v;        r = u%v;        j = i-q*j;        i = k;        k = j;    x = c*k;    y = c*(1-a*k)/b;    print(x, y);

本节到此结束，欲知后事如何，请看下回分解。