149 lines
4.5 KiB
Markdown
149 lines
4.5 KiB
Markdown
---
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id: 598eea87e5cf4b116c3ff81a
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title: Factors of a Mersenne number
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challengeType: 5
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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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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>p</sup>-1</code>.
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# --hints--
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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)` 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)` should return the string `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)` should return the string `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)` should return the string `M929 = 2^929-1 is composite with factor 13007`.
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```js
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assert.equal(
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check_mersenne(929),
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'M929 = 2^929-1 is composite with factor 13007'
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);
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```
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# --seed--
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## --seed-contents--
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```js
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function check_mersenne(p) {
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}
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```
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# --solutions--
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```js
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function check_mersenne(p){
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function isPrime(value){
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for (let i=2; i < value; i++){
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if (value % i == 0){
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return false;
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}
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if (value % i != 0){
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return true;
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}
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}
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}
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function trial_factor(base, exp, mod){
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let square, bits;
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square = 1;
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bits = exp.toString(2).split('');
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for (let i=0,ln=bits.length; i<ln; i++){
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square = Math.pow(square, 2) * (bits[i] == 1 ? base : 1) % mod;
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}
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return (square == 1);
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}
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function mersenne_factor(p){
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let limit, k, q;
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limit = Math.sqrt(Math.pow(2,p) - 1);
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k = 1;
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while ((2*k*p - 1) < limit){
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q = 2*k*p + 1;
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if (isPrime(q) && (q % 8 == 1 || q % 8 == 7) && trial_factor(2,p,q)){
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return q; // q is a factor of 2**p-1
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}
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k++;
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}
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return null;
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}
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let f, result;
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result="M"+p+" = 2^"+p+"-1 is ";
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f = mersenne_factor(p);
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result+=f == null ? "prime" : "composite with factor "+f;
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return result;
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}
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```
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