For two natural numbers 1≤a≤b1≤a≤b define the graph Ga,bGa,b​ as follows: its vertex set is {a,a+1,…,b}{a,a+1,…,b}, and its edge set consists of all pairs {i,j}{i,j} such that i≠ji​=j and i+ji+j is a prime number.

**How many edges does** G1,6G1,6​ **have?**

We are tasked with determining how many edges the graph \( G_{1,6} \) has. 

### Step 1: Understanding the vertex set and edge conditions
The vertex set of \( G_{1,6} \) is:

\[
\{1, 2, 3, 4, 5, 6\}
\]

The edges in the graph are defined by pairs of vertices \( \{i, j\} \) such that \( i \neq j \) and \( i + j \) is a prime number.

### Step 2: Calculate sums and check for primality
We now check the pairs of vertices and compute their sums to determine which are prime:

- \( i = 1 \):
  - \( 1 + 2 = 3 \) (prime)
  - \( 1 + 3 = 4 \) (not prime)
  - \( 1 + 4 = 5 \) (prime)
  - \( 1 + 5 = 6 \) (not prime)
  - \( 1 + 6 = 7 \) (prime)

- \( i = 2 \):
  - \( 2 + 3 = 5 \) (prime)
  - \( 2 + 4 = 6 \) (not prime)
  - \( 2 + 5 = 7 \) (prime)
  - \( 2 + 6 = 8 \) (not prime)

- \( i = 3 \):
  - \( 3 + 4 = 7 \) (prime)
  - \( 3 + 5 = 8 \) (not prime)
  - \( 3 + 6 = 9 \) (not prime)

- \( i = 4 \):
  - \( 4 + 5 = 9 \) (not prime)
  - \( 4 + 6 = 10 \) (not prime)

- \( i = 5 \):
  - \( 5 + 6 = 11 \) (prime)

### Step 3: List all valid edges
The valid edges correspond to the sums that resulted in a prime number:

- \( \{1, 2\}, \{1, 4\}, \{1, 6\} \)
- \( \{2, 3\}, \{2, 5\} \)
- \( \{3, 4\} \)
- \( \{5, 6\} \)

### Step 4: Count the edges
There are 7 edges in total.

Thus, the graph \( G_{1,6} \) has **7 edges**.

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Do you want more details on how the sums were checked for primality, or do you have any questions?

### 5 Related Questions:
1. How do we efficiently check whether a number is prime?
2. What are the properties of a graph with vertices defined by prime-related conditions?
3. Can we generalize this problem for larger \( b \) values?
4. How does this graph compare to a complete graph \( K_6 \)?
5. What is the maximum number of edges in a graph where sums of vertices must be prime?

**Tip:** For small numbers, checking whether a number is prime can be done by testing divisibility up to the square root of the number.

For two natural numbers 1≤a≤b, define the graph Ga,b as follows: its vertex set is {a,a+1,…,b}, and its edge set consists of all pairs {i,j} such that i≠j and i+j is a prime number. How many edges does G1,6 have?

Determine the number of edges in the graph G1,6 by analyzing the sums of vertex pairs and checking for primality. This problem integrates concepts from graph theory and number theory, making it suitable for students in Grades 10-12.

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