> For the complete documentation index, see [llms.txt](https://docs.1l.lol/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.1l.lol/number-theory/prime-test.md).

# 素数判定（试除法）

### 素数的定义

大于 1 且只有 1 和它本身两个正因数的数。**1 不是素数**，2 是唯一的偶素数。

### 试除法 O(√n)

若 n 有大于 1 的因子，必有一个不超过 `√n`：

```cpp
bool isPrime(int n) {
    if (n < 2) return false;
    for (int i = 2; i * i <= n; i++)
        if (n % i == 0) return false;
    return true;
}
```

### 优化

* 循环条件写 `i * i <= n`，避免开方误差，复杂度 O(√n)。
* 先判断 2，再只试奇数因子，可再快一倍：

```cpp
bool isPrime(int n) {
    if (n < 2) return false;
    if (n == 2) return true;
    if (n % 2 == 0) return false;
    for (int i = 3; i * i <= n; i += 2)
        if (n % i == 0) return false;
    return true;
}
```

### 复杂度与适用

* 单次判断 O(√n)。n 达 10¹² 也能快速判断。
* 若要判断很多个数或求素数表，应使用**埃氏筛法 / 线性筛法**。

### 常见错误

* 忘记判断 `n < 2`，把 1 或负数当成素数。
* `i <= n` 而不是 `i * i <= n`，复杂度过高。


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