Prime Graph

题意翻译

###### 给出n个点，求一个图。 ###### 要求这张图满足： 1. 图是简单无向图（即没有重边和自环） 2. 点的编号为1~n 3. 图的边数是素数 4. 每个点的度都是素数 （0,1不是素数） ###### 注意：图可以不连通，给出任意一张符合上述要求的图即可。

题目描述

Every person likes prime numbers. Alice is a person, thus she also shares the love for them. Bob wanted to give her an affectionate gift but couldn't think of anything inventive. Hence, he will be giving her a graph. How original, Bob! Alice will surely be thrilled! When building the graph, he needs four conditions to be satisfied: - It must be a simple undirected graph, i.e. without multiple (parallel) edges and self-loops. - The number of vertices must be exactly $ n $ — a number he selected. This number is not necessarily prime. - The total number of edges must be prime. - The degree (i.e. the number of edges connected to the vertex) of each vertex must be prime. Below is an example for $ n = 4 $ . The first graph (left one) is invalid as the degree of vertex $ 2 $ (and $ 4 $ ) equals to $ 1 $ , which is not prime. The second graph (middle one) is invalid as the total number of edges is $ 4 $ , which is not a prime number. The third graph (right one) is a valid answer for $ n = 4 $ . Note that the graph can be disconnected. Please help Bob to find any such graph!

输入输出格式

输入格式

The input consists of a single integer $ n $ ( $ 3 \leq n \leq 1\,000 $ ) — the number of vertices.

输出格式

If there is no graph satisfying the conditions, print a single line containing the integer $ -1 $ . Otherwise, first print a line containing a prime number $ m $ ( $ 2 \leq m \leq \frac{n(n-1)}{2} $ ) — the number of edges in the graph. Then, print $ m $ lines, the $ i $ -th of which containing two integers $ u_i $ , $ v_i $ ( $ 1 \leq u_i, v_i \leq n $ ) — meaning that there is an edge between vertices $ u_i $ and $ v_i $ . The degree of each vertex must be prime. There must be no multiple (parallel) edges or self-loops. If there are multiple solutions, you may print any of them. Note that the graph can be disconnected.

输入输出样例

输入样例 #1

4

输出样例 #1

5
1 2
1 3
2 3
2 4
3 4

输入样例 #2

8

输出样例 #2

13
1 2
1 3
2 3
1 4
2 4
1 5
2 5
1 6
2 6
1 7
1 8
5 8
7 8

说明

The first example was described in the statement. In the second example, the degrees of vertices are $ [7, 5, 2, 2, 3, 2, 2, 3] $ . Each of these numbers is prime. Additionally, the number of edges, $ 13 $ , is also a prime number, hence both conditions are satisfied.