chore(i8n,learn): processed translations
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Mrugesh Mohapatra
Mrugesh Mohapatra
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---
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id: 598eea87e5cf4b116c3ff81a
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title: 梅森数的因素
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title: Factors of a Mersenne number
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challengeType: 5
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videoUrl: ''
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forumTopicId: 302264
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dashedName: factors-of-a-mersenne-number
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---
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# --description--
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<p>梅森数是2 <sup>P</sup> -1形式的数字。 </p><p>如果P是素数，那么梅森数可能是梅森素数</p><p> （如果P不是素数，则梅森数也不是素数）。 </p><p>在搜索梅森素数时，通过在开始可能冗长的<a href='http://rosettacode.org/wiki/Lucas-Lehmer test' title='Lucas-Lehmer测试'>Lucas-Lehmer检验</a>之前找到一个小因子来消除指数是有利的。 </p><p>有非常有效的算法来确定数字是否除以2 <sup>P</sup> -1（或等效地，如果2 <sup>P</sup> mod（数字）= 1）。 </p><p>有些语言已经有了这个exponent-and-mod操作的内置实现（称为modPow或类似的）。 </p><p>以下是如何自己实现这个modPow： </p><p>例如，让我们计算2 <sup>23</sup> mod 47。 </p><p>将指数23转换为二进制，得到10111.从<tt>square</tt> = 1开始，重复平方。 </p><p>卸下指数的最高位，并且如果它是1 <tt>平方</tt>乘以由所述幂（2）的基础上，然后计算<tt>平方</tt>模47。 </p><p>在下一步中使用最后一步的模数结果作为<tt>square</tt>的初始值： </p><p>删除可选</p><p>方形顶部位乘以2 mod 47 </p><p> ------------ ------- ------------- ------ </p><p> 1 * 1 = 1 1 0111 1 * 2 = 2 2 </p><p> 2 * 2 = 4 0 111否4 </p><p> 4 * 4 = 16 1 11 16 * 2 = 32 32 </p><p> 32 * 32 = 1024 1 1 1024 * 2 = 2048 27 </p><p> 27 * 27 = 729 1 729 * 2 = 1458 1 </p><p>由于2 <sup>23</sup> mod 47 = 1,47是2 <sup>P</sup> -1的因子。 </p><p> （要看到这一点，从两边减去1：2 <sup>23</sup> -1 = 0 mod 47.） </p><p>由于我们已经证明47是一个因子，因此2 <sup>23</sup> -1不是素数。 </p><p> Mersenne数字的其他属性使我们能够进一步完善这一过程。 </p><p>任何因子q为2 <sup>P</sup> -1必须是2kP + 1的形式，k是正整数或零。此外，q必须是1或7 mod 8。 </p><p>最后任何潜在因素q必须是<a href='http://rosettacode.org/wiki/Primality by Trial Division' title='审判分庭的原始性'>素数</a> 。 </p><p>与其他试验划分算法一样，算法在2kP + 1> sqrt（N）时停止。 </p><p>这些素性测试仅适用于P为素数的Mersenne数。例如，M <sub>4</sub> = 15不使用这些技术产生因子，而是产生3和5的因子，两者都不符合2kP + 1。 </p>任务： <p>使用上述方法找到因子2 <sup>929</sup> -1（又名M929） </p>相关任务： <a href='http://rosettacode.org/wiki/count in factors' title='算上因素'>计数因素</a> <a href='http://rosettacode.org/wiki/prime decomposition' title='主要分解'>素数分解</a> <a href='http://rosettacode.org/wiki/factors of an integer' title='整数的因子'>的整数的因素</a> <a href='http://rosettacode.org/wiki/Sieve of Eratosthenes' title='Eratosthenes的筛子'>埃拉托塞尼的筛</a> <a href='http://rosettacode.org/wiki/primality by trial division' title='审判分裂的素性'>通过试验除法素性</a> <a href='http://rosettacode.org/wiki/trial factoring of a Mersenne number' title='试用Mensenne数的因式'>梅森数的试验理</a> <a href='http://rosettacode.org/wiki/partition an integer X into N primes' title='将整数X划分为N个素数'>分区的整数X为N个素数</a> <a href='http://rosettacode.org/wiki/sequence of primes by Trial Division' title='审判分庭的素数序列'>由审判庭素数的序列</a> <a href='https://www.youtube.com/watch?v=SNwvJ7psoow' title='链接：https：//www.youtube.com/watch？v = SNwvJ7psoow'>在1948年计算机：2¹²⁷-1</a>
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A Mersenne number is a number in the form of <code>2<sup>P</sup>-1</code>.
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If `P` is prime, the Mersenne number may be a Mersenne prime. (If `P` is not prime, the Mersenne number is also not prime.)
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In the search for Mersenne prime numbers it is advantageous to eliminate exponents by finding a small factor before starting a, potentially lengthy, [Lucas-Lehmer test](https://rosettacode.org/wiki/Lucas-Lehmer test "Lucas-Lehmer test").
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There are very efficient algorithms for determining if a number divides <code>2<sup>P</sup>-1</code> (or equivalently, if <code>2<sup>P</sup> mod (the number) = 1</code>).
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Some languages already have built-in implementations of this exponent-and-mod operation (called modPow or similar).
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The following is how to implement this modPow yourself:
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For example, let's compute <code>2<sup>23</sup> mod 47</code>.
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Convert the exponent 23 to binary, you get 10111. Starting with <code><tt>square</tt> = 1</code>, repeatedly square it.
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Remove the top bit of the exponent, and if it's 1 multiply `square` by the base of the exponentiation (2), then compute <code><tt>square</tt> modulo 47</code>.
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Use the result of the modulo from the last step as the initial value of `square` in the next step:
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<pre>Remove Optional
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square top bit multiply by 2 mod 47
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------------ ------- ------------- ------
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1*1 = 1 1 0111 1*2 = 2 2
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2*2 = 4 0 111 no 4
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4*4 = 16 1 11 16*2 = 32 32
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32*32 = 1024 1 1 1024*2 = 2048 27
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27*27 = 729 1 729*2 = 1458 1
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</pre>
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Since <code>2<sup>23</sup> mod 47 = 1</code>, 47 is a factor of <code>2<sup>P</sup>-1</code>.
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(To see this, subtract 1 from both sides: <code>2<sup>23</sup>-1 = 0 mod 47</code>.)
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Since we've shown that 47 is a factor, <code>2<sup>23</sup>-1</code> is not prime.
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Further properties of Mersenne numbers allow us to refine the process even more.
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Any factor `q` of <code>2<sup>P</sup>-1</code> must be of the form `2kP+1`, `k` being a positive integer or zero. Furthermore, `q` must be `1` or `7 mod 8`.
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Finally any potential factor `q` must be [prime](https://rosettacode.org/wiki/Primality by Trial Division "Primality by Trial Division").
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As in other trial division algorithms, the algorithm stops when `2kP+1 > sqrt(N)`.These primarily tests only work on Mersenne numbers where `P` is prime. For example, <code>M<sub>4</sub>=15</code> yields no factors using these techniques, but factors into 3 and 5, neither of which fit `2kP+1`.
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# --instructions--
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Using the above method find a factor of <code>2<sup>929</sup>-1</code> (aka M929)
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# --hints--
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`check_mersenne`是一个函数。
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`check_mersenne` should be a function.
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```js
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assert(typeof check_mersenne === 'function');
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```
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`check_mersenne(3)`应该返回一个字符串。
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`check_mersenne(3)` should return a string.
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```js
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assert(typeof check_mersenne(3) == 'string');
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```
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`check_mersenne(3)`应该返回“M3 = 2 ^ 3-1是素数”。
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`check_mersenne(3)` should return "M3 = 2^3-1 is prime".
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```js
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assert.equal(check_mersenne(3), 'M3 = 2^3-1 is prime');
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```
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`check_mersenne(23)`应返回“M23 = 2 ^ 23-1与因子47复合”。
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`check_mersenne(23)` should return "M23 = 2^23-1 is composite with factor 47".
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```js
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assert.equal(check_mersenne(23), 'M23 = 2^23-1 is composite with factor 47');
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```
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`check_mersenne(929)`应返回“M929 = 2 ^ 929-1与因子13007复合
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`check_mersenne(929)` should return "M929 = 2^929-1 is composite with factor 13007
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```js
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assert.equal(
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