Friday, April 8, 2016

Super Cube - Twitter


https://www.quora.com/How-can-the-pair-a-b-a-b-in-mathbb-Z-be-found-such-that-a-frac-1-3-+-b-frac-1-3-3-is-also-an-integer

How can the pair (a,b)abZ be found such that (a13+b13)3 is also an integer?

http://www.1point3acres.com/bbs/forum.php?mod=viewthread&tid=180327
第一道是给两个数M和N,求从1到M和1到N之间的数有哪些组合满足两个数立方根的和的三次方是整数
One can easily observe that this is satisfied whenever a and b are themselves cube of integers. For example if ab=8, 27; then the expression becomes 125. However this can be generalised to the case where the cube free part of a and b are equal.
Lets define the cube-free part of a given integer N. We first prime factorise the integer. Then we divide N by all cubes (or 6th powers etc.) of primes that occur at least 3 times (or 6 times etc.) in the factorisation. So for 54=33×2 the cube free part is 2. Hence the pair 2, 54 would also give an integer in the final expression. It is indeed 128. Now the first case is just a specialisation of this when the cube free part of both integers are 1.

第一题先把式子展开,发现只要找到a^(1/3), b^(1/3), a^(2/3), b^(2/3)(其实也就是a^(1/3), b^(1/3))
感觉这是充分条件。比如3和24,立方根和的立方也是整数

http://www.1point3acres.com/bbs/forum.php?mod=viewthread&tid=180327&page=1#pid2336923
大概的思路就是把式子展开发现只要a^(1 / 3) * b^(2 / 3)和a^(2 / 3) * b^(1 / 3)为整数就能保证结果为整数,如果a^(1 / 3) = b^(1 / 3) * X (X为整数)和b(1 / 3) = a ^ (1 / 3) * X (X为整数),那么a^(1 / 3) * b^(2 / 3) = b * X和a^(2 / 3) * b^(1 / 3) = a * X就肯定是整数,即找到满足a = b * X ^ 3和b = a * X ^ 3的所有a,b组合
  1. public List<int[]> cubeRoot(int m, int n) {
  2.                 List<int[]> res = new ArrayList<int[]>();
  3.                 List<Integer> listM = new ArrayList<Integer>();
  4.                 List<Integer> listN = new ArrayList<Integer>();
  5.                 int cur = 1;
  6.                 while(cur <= m / cur / cur) {
  7.                         listM.add(cur * cur * cur);. more info on 1point3acres.com
  8.                         cur++;. 1point3acres.com/bbs
  9.                 }.1point3acres缃�
  10.                 cur = 1;
  11.                 while(cur <= n / cur / cur) {
  12.                         listN.add(cur * cur * cur);
  13.                         cur++;. from: 1point3acres.com/bbs 
  14.                 }
  15.                 int a = 1;
  16.                 int b = 1;
  17.                 HashMap<Integer, HashSet<Integer>> map = new HashMap<Integer, HashSet<Integer>>();
  18.                 while(b <= n) {
  19.                         for(int numA : listM) {
  20.                                 a = b * numA;
  21.                                 if(a <= m) {
  22.                                         if(!map.containsKey(a)) {
  23.                                                 map.put(a, new HashSet<Integer>());
  24.                                         }
  25.                                         map.get(a).add(b);. 1point 3acres 璁哄潧
  26.                                         res.add(new int[]{a, b});
  27.                                 }
  28.                                 else {
  29.                                         break;
  30.                                 }
  31.                         }.1point3acres缃�
  32.                         b++;. From 1point 3acres bbs
  33.                 }.鐣欏璁哄潧-涓€浜�-涓夊垎鍦�
  34.                 a = 1;
  35.                 b = 1;
  36.                 while(a <= m) {
  37.                         for(int numB : listN) {
  38.                                 b = a * numB;
  39.                                 if(b <= n) {
  40.                                         if(!map.containsKey(a)) {
  41.                                                 map.put(a, new HashSet<Integer>());
  42.                                         }
  43.                                         else {
  44.                                                 if(map.get(a).contains(b)) {
  45.                                                         continue;
  46.                                                 }
  47.                                         }
  48.                                         map.get(a).add(b);
  49.                                         res.add(new int[]{a, b});
  50.                                 }
  51.                                 else {
  52.                                         break;
  53.                                 }
  54.                         }
  55.                         a++;
  56.                 }. 涓€浜�-涓夊垎-鍦帮紝鐙鍙戝竷
  57.                 return res;. more info on 1point3acres.com
  58.         }
http://www.1point3acres.com/bbs/thread-181810-1-1.html
public static int cubes(int M, int N){
int count = 1;
for(int a = 1; a <= M; a++){
for(int b = 1; b <= N; b++){
double x = Math.pow(a, 1/3.) + Math.pow(b, 1/3.);
if(x == Math.round(x)) count++;
}
}
return count;
}


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