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+# How to find prime Numbers efficiently
+
+**Translator: [shazi4399](https://github.com/shazi4399)**
+
+**Author: [labuladong](https://github.com/labuladong)**
+
+The definition of a prime number seems simple,which is said to be prime number if it can be divided by 1 and itself.
+
+However,don't think that the definition of prime numbers is simple. I am afraid that few people can write a prime-related algorithm that works really efficiently. Let's say you write a function like this:
+
+```java
+// Returns several primes in the interval [2, n) 
+int countPrimes(int n)
+
+// E.g. countPrimes (10) returns 4
+// Because 2,3,5,7 is prime numbers
+```
+
+How would you progrma this function? I think you maybe write like this:
+
+```java
+int countPrimes(int n) {
+    int count = 0;
+    for (int i = 2; i < n; i++)
+        if (isPrim(i)) count++;
+    return count;
+}
+
+// Determines whether integer n is prime
+boolean isPrime(int n) {
+    for (int i = 2; i < n; i++)
+        if (n % i == 0)
+            // There are other divisibility factors
+            return false;
+    return true;
+}
+```
+
+The time complexity is O (n ^ 2), which is a big problem.**First of all, the idea of using the isPrime function to assist is not efficient; and even if you want to use the isPrime function, there is computational redundancy in writing the algorithm**.
+
+Let's briefly talk about **how to write an algorithm if you want to determine whether a number is prime or not**. Just slightly modify the for loop condition in the isPrim code above:
+
+```java
+boolean isPrime(int n) {
+    for (int i = 2; i * i <= n; i++)
+        ...
+}
+```
+
+In other words, `i` does not need to traverse to` n`, but only to `sqrt (n)`. Why? let's take an example, suppose `n = 12`.
+
+```java
+12 = 2 × 6
+12 = 3 × 4
+12 = sqrt(12) × sqrt(12)
+12 = 4 × 3
+12 = 6 × 2
+```
+
+As you can see, the last two products are the reverse of the previous two, and the critical point of inversion is at `sqrt (n)`.
+
+In other words, if no divisible factor is found within the interval `[[2, sqrt (n)]`, you can directly conclude that `n` is a prime number, because in the interval `[[sqrt (n), n] ` Nor will you find a divisible factor.
+
+Now, the time complexity of the `isPrime` function is reduced to O (sqrt (N)), ** but we don't actually need this function to implement the` countPrimes` function. The above just hope that readers understand the meaning of `sqrt (n)`, because it will be used again later.
+
+
+### Efficient implementation `countPrimes`
+
+The core idea of efficiently solving this problem is to reverse the conventional idea above:
+
+First from 2, we know that 2 is a prime number, then 2 × 2 = 4, 3 × 2 = 6, 4 × 2 = 8 ... all are not prime numbers.
+
+Then we found that 3 is also a prime number, so 3 × 2 = 6, 3 × 3 = 9, 3 × 4 = 12 ... are also impossible to be prime numbers.
+
+Seeing this, do you understand the logic of this exclusion method a bit? First look at our first version of the code:
+
+```java
+int countPrimes(int n) {
+    boolean[] isPrim = new boolean[n];
+    // Initialize the arrays to true
+    Arrays.fill(isPrim, true);
+
+    for (int i = 2; i < n; i++) 
+        if (isPrim[i]) 
+            // Multiples of i cannot be prime
+            for (int j = 2 * i; j < n; j += i) 
+                    isPrim[j] = false;
+    
+    int count = 0;
+    for (int i = 2; i < n; i++)
+        if (isPrim[i]) count++;
+    
+    return count;
+}
+```
+
+If you can understand the above code, then you have mastered the overall idea, but there are two subtle areas that can be optimized.
+
+First of all, recall the `isPrime` function that just judges whether a number is prime. Due to the symmetry of the factors, the for loop only needs to traverse` [2, sqrt (n)] `. Here is similar, our outer for loop only needs to traverse to `sqrt (n)`:
+
+```java
+for (int i = 2; i * i < n; i++) 
+    if (isPrim[i]) 
+        ...
+```
+
+In addition, it is difficult to notice that the inner for loop can also be optimized. Our previous approach was:
+
+```java
+for (int j = 2 * i; j < n; j += i) 
+    isPrim[j] = false;
+```
+
+This can mark all integer multiples of `i` as` false`, but there is still computational redundancy.
+
+For example, when `n = 25` and` i = 4`, the algorithm will mark numbers such as 4 × 2 = 8, 4 × 3 = 12, and so on, but these two numbers have been marked by 2 × 4 and 3 × 4 that is `i = 2` and` i = 3`.
+
+We can optimize it slightly so that `j` traverses from the square of` i` instead of starting from `2 * i`:
+
+```java
+for (int j = i * i; j < n; j += i) 
+    isPrim[j] = false;
+```
+
+In this way, the algorithm for counting prime numbers is efficiently implemented. In fact, this algorithm has a name, which called Sieve of Eratosthenes. Take a look at the complete final code:
+
+```java
+int countPrimes(int n) {
+    boolean[] isPrim = new boolean[n];
+    Arrays.fill(isPrim, true);
+    for (int i = 2; i * i < n; i++) 
+        if (isPrim[i]) 
+            for (int j = i * i; j < n; j += i) 
+                isPrim[j] = false;
+    
+    int count = 0;
+    for (int i = 2; i < n; i++)
+        if (isPrim[i]) count++;
+    
+    return count;
+}
+```
+
+**The time complexity of this algorithm is difficult to calculate**.It is obvious that the time is related to these two nested for loops. The operands should be:
+
+  n/2 + n/3 + n/5 + n/7 + ...
+= n × (1/2 + 1/3 + 1/5 + 1/7...)
+
+In parentheses, ther is the inverse of the prime number .The final result is  O(N * loglogN),and readers interested in this can refer to the time complexity of the algorithm
+
+That is all about how to find prime Numbers.The seemingly simple problem does has a lot of details to polish
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-# 如何高效寻找素数
-
-素数的定义看起来很简单，如果一个数如果只能被 1 和它本身整除，那么这个数就是素数。
-
-不要觉得素数的定义简单，恐怕没多少人真的能把素数相关的算法写得高效。比如让你写这样一个函数：
-
-```java
-// 返回区间 [2, n) 中有几个素数 
-int countPrimes(int n)
-
-// 比如 countPrimes(10) 返回 4
-// 因为 2,3,5,7 是素数
-```
-
-你会如何写这个函数？我想大家应该会这样写：
-
-```java
-int countPrimes(int n) {
-    int count = 0;
-    for (int i = 2; i < n; i++)
-        if (isPrim(i)) count++;
-    return count;
-}
-
-// 判断整数 n 是否是素数
-boolean isPrime(int n) {
-    for (int i = 2; i < n; i++)
-        if (n % i == 0)
-            // 有其他整除因子
-            return false;
-    return true;
-}
-```
-
-这样写的话时间复杂度 O(n^2)，问题很大。**首先你用 isPrime 函数来辅助的思路就不够高效；而且就算你要用 isPrime 函数，这样写算法也是存在计算冗余的**。
-
-先来简单说下**如果你要判断一个数是不是素数，应该如何写算法**。只需稍微修改一下上面的 isPrim 代码中的 for 循环条件：
-
-```java
-boolean isPrime(int n) {
-    for (int i = 2; i * i <= n; i++)
-        ...
-}
-```
-
-换句话说，`i` 不需要遍历到 `n`，而只需要到 `sqrt(n)` 即可。为什么呢，我们举个例子，假设 `n = 12`。
-
-```java
-12 = 2 × 6
-12 = 3 × 4
-12 = sqrt(12) × sqrt(12)
-12 = 4 × 3
-12 = 6 × 2
-```
-
-可以看到，后两个乘积就是前面两个反过来，反转临界点就在 `sqrt(n)`。
-
-换句话说，如果在 `[2,sqrt(n)]` 这个区间之内没有发现可整除因子，就可以直接断定 `n` 是素数了，因为在区间 `[sqrt(n),n]` 也一定不会发现可整除因子。
-
-现在，`isPrime` 函数的时间复杂度降为 O(sqrt(N))，**但是我们实现 `countPrimes` 函数其实并不需要这个函数**，以上只是希望读者明白 `sqrt(n)` 的含义，因为等会还会用到。
-
-
-### 高效实现 `countPrimes`
-
-高效解决这个问题的核心思路是和上面的常规思路反着来：
-
-首先从 2 开始，我们知道 2 是一个素数，那么 2 × 2 = 4, 3 × 2 = 6, 4 × 2 = 8... 都不可能是素数了。
-
-然后我们发现 3 也是素数，那么 3 × 2 = 6, 3 × 3 = 9, 3 × 4 = 12... 也都不可能是素数了。
-
-看到这里，你是否有点明白这个排除法的逻辑了呢？先看我们的第一版代码：
-
-```java
-int countPrimes(int n) {
-    boolean[] isPrim = new boolean[n];
-    // 将数组都初始化为 true
-    Arrays.fill(isPrim, true);
-
-    for (int i = 2; i < n; i++) 
-        if (isPrim[i]) 
-            // i 的倍数不可能是素数了
-            for (int j = 2 * i; j < n; j += i) 
-                    isPrim[j] = false;
-    
-    int count = 0;
-    for (int i = 2; i < n; i++)
-        if (isPrim[i]) count++;
-    
-    return count;
-}
-```
-
-如果上面这段代码你能够理解，那么你已经掌握了整体思路，但是还有两个细微的地方可以优化。
-
-首先，回想刚才判断一个数是否是素数的 `isPrime` 函数，由于因子的对称性，其中的 for 循环只需要遍历 `[2,sqrt(n)]` 就够了。这里也是类似的，我们外层的 for 循环也只需要遍历到 `sqrt(n)`：
-
-```java
-for (int i = 2; i * i < n; i++) 
-    if (isPrim[i]) 
-        ...
-```
-
-除此之外，很难注意到内层的 for 循环也可以优化。我们之前的做法是：
-
-```java
-for (int j = 2 * i; j < n; j += i) 
-    isPrim[j] = false;
-```
-
-这样可以把 `i` 的整数倍都标记为 `false`，但是仍然存在计算冗余。
-
-比如 `n = 25`，`i = 4` 时算法会标记 4 × 2 = 8，4 × 3 = 12 等等数字，但是这两个数字已经被 `i = 2` 和 `i = 3` 的 2 × 4 和 3 × 4 标记了。
-
-我们可以稍微优化一下，让 `j` 从 `i` 的平方开始遍历，而不是从 `2 * i` 开始：
-
-```java
-for (int j = i * i; j < n; j += i) 
-    isPrim[j] = false;
-```
-
-这样，素数计数的算法就高效实现了，其实这个算法有一个名字，叫做 Sieve of Eratosthenes。看下完整的最终代码：
-
-```java
-int countPrimes(int n) {
-    boolean[] isPrim = new boolean[n];
-    Arrays.fill(isPrim, true);
-    for (int i = 2; i * i < n; i++) 
-        if (isPrim[i]) 
-            for (int j = i * i; j < n; j += i) 
-                isPrim[j] = false;
-    
-    int count = 0;
-    for (int i = 2; i < n; i++)
-        if (isPrim[i]) count++;
-    
-    return count;
-}
-```
-
-**该算法的时间复杂度比较难算**，显然时间跟这两个嵌套的 for 循环有关，其操作数应该是：
-
-  n/2 + n/3 + n/5 + n/7 + ...
-= n × (1/2 + 1/3 + 1/5 + 1/7...)
-
-括号中是素数的倒数。其最终结果是 O(N * loglogN)，有兴趣的读者可以查一下该算法的时间复杂度证明。
-
-以上就是素数算法相关的全部内容。怎么样，是不是看似简单的问题却有不少细节可以打磨呀？
-
-**致力于把算法讲清楚！欢迎关注我的微信公众号 labuladong，查看更多通俗易懂的文章**：
-
-![labuladong](../pictures/labuladong.png)
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