因子循环变为0

Question

我运行了一个简单的程序,使用编译语言,使用两个简单的循环计算前几个自然数的阶乘,一个外部跟踪我们计算阶乘的数字和一个计算阶乘的内部数.将每个自然数从1乘以数字本身.该程序对于第一个自然数是完美的,然后大约从第13个值开始计算的因子显然是错误的.这是由于在现代计算机中实现的整数运算,我可以理解为什么会出现负值.我不明白的原因是,为什么,这是我在不同的计算机上测试过的,经过非常少量的因子计算它总是达到数字零.当然,如果第n个阶乘被评估为0,那么第(n + 1)个阶乘也将被评估为0,依此类推,但为什么数字0总是出现在非常少量的阶乘之后阶乘计算？

编辑:你可能想知道为什么我使用两个不同的周期而不是只有一个...我这样做是为了强迫计算机从头开始重新计算每个因子,只是为了测试这个因素确实是因子总是为0而且它是不是偶然的.

这是我的输出:

Answer 1

从34!开始,所有阶乘都可以被2 ^ 32整除.因此,当您的计算机程序计算结果模2 ^ 32(虽然您没有说明您正在使用的编程语言,这很可能),但结果始终为0.

这是一个在Python中计算因子mod 2 ^ 32的程序:

def sint(r):
    r %= (1 << 32)
    return r if r < (1 << 31) else r - (1 << 32)

r = 1
for i in xrange(1, 40):
    r *= i
    print '%d! = %d mod 2^32' % (i, sint(r))

这给出了这个输出,它与你自己的程序的输出一致:

1! = 1 mod 2^32
2! = 2 mod 2^32
3! = 6 mod 2^32
4! = 24 mod 2^32
5! = 120 mod 2^32
6! = 720 mod 2^32
7! = 5040 mod 2^32
8! = 40320 mod 2^32
9! = 362880 mod 2^32
10! = 3628800 mod 2^32
11! = 39916800 mod 2^32
12! = 479001600 mod 2^32
13! = 1932053504 mod 2^32
14! = 1278945280 mod 2^32
15! = 2004310016 mod 2^32
16! = 2004189184 mod 2^32
17! = -288522240 mod 2^32
18! = -898433024 mod 2^32
19! = 109641728 mod 2^32
20! = -2102132736 mod 2^32
21! = -1195114496 mod 2^32
22! = -522715136 mod 2^32
23! = 862453760 mod 2^32
24! = -775946240 mod 2^32
25! = 2076180480 mod 2^32
26! = -1853882368 mod 2^32
27! = 1484783616 mod 2^32
28! = -1375731712 mod 2^32
29! = -1241513984 mod 2^32
30! = 1409286144 mod 2^32
31! = 738197504 mod 2^32
32! = -2147483648 mod 2^32
33! = -2147483648 mod 2^32
34! = 0 mod 2^32
35! = 0 mod 2^32
36! = 0 mod 2^32
37! = 0 mod 2^32
38! = 0 mod 2^32
39! = 0 mod 2^32

这里是这个阶乘系数的精确值的表格,显示每个阶乘2的幂数:

1! = 1. Divisible by 2^0
2! = 2. Divisible by 2^1
3! = 6. Divisible by 2^1
4! = 24. Divisible by 2^3
5! = 120. Divisible by 2^3
6! = 720. Divisible by 2^4
7! = 5040. Divisible by 2^4
8! = 40320. Divisible by 2^7
9! = 362880. Divisible by 2^7
10! = 3628800. Divisible by 2^8
11! = 39916800. Divisible by 2^8
12! = 479001600. Divisible by 2^10
13! = 6227020800. Divisible by 2^10
14! = 87178291200. Divisible by 2^11
15! = 1307674368000. Divisible by 2^11
16! = 20922789888000. Divisible by 2^15
17! = 355687428096000. Divisible by 2^15
18! = 6402373705728000. Divisible by 2^16
19! = 121645100408832000. Divisible by 2^16
20! = 2432902008176640000. Divisible by 2^18
21! = 51090942171709440000. Divisible by 2^18
22! = 1124000727777607680000. Divisible by 2^19
23! = 25852016738884976640000. Divisible by 2^19
24! = 620448401733239439360000. Divisible by 2^22
25! = 15511210043330985984000000. Divisible by 2^22
26! = 403291461126605635584000000. Divisible by 2^23
27! = 10888869450418352160768000000. Divisible by 2^23
28! = 304888344611713860501504000000. Divisible by 2^25
29! = 8841761993739701954543616000000. Divisible by 2^25
30! = 265252859812191058636308480000000. Divisible by 2^26
31! = 8222838654177922817725562880000000. Divisible by 2^26
32! = 263130836933693530167218012160000000. Divisible by 2^31
33! = 8683317618811886495518194401280000000. Divisible by 2^31
34! = 295232799039604140847618609643520000000. Divisible by 2^32
35! = 10333147966386144929666651337523200000000. Divisible by 2^32
36! = 371993326789901217467999448150835200000000. Divisible by 2^34
37! = 13763753091226345046315979581580902400000000. Divisible by 2^34
38! = 523022617466601111760007224100074291200000000. Divisible by 2^35
39! = 20397882081197443358640281739902897356800000000. Divisible by 2^35