To determine how many triplets of prime numbers can be formed such that the terms of the triplet are in Arithmetic Progression (A.P.), we can follow these steps:
### Step 1: Understanding the Condition for A.P.
In an A.P., the middle term is the average of the other two terms. For three numbers \( a, b, c \) to be in A.P., the condition is:
\[ b - a = c - b \]
This can be rearranged to give:
\[ 2b = a + c \]
This means that \( b \) must be the average of \( a \) and \( c \).
### Step 2: Identifying the Prime Numbers
We need to consider prime numbers. The first few prime numbers are:
\[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541 \]
### Step 3: Finding Triplets
We need to find triplets \( (p_1, p_2, p_3) \) such that:
1. \( p_1, p_2, p_3 \) are all prime.
2. They satisfy the A.P. condition.
### Step 4: Analyzing the Triplets
1. **Odd and Even Primes**: The only even prime number is 2. If we include 2 in the triplet, the other two primes must be odd (since the sum of an even and two odds is odd, which is not prime). Thus, we can form triplets like \( (2, p_2, p_3) \) where \( p_2 \) and \( p_3 \) are odd primes.
2. **All Odd Primes**: If all three primes are odd, then they can be in A.P. as well. For example, triplets like \( (3, 5, 7) \), \( (5, 7, 11) \), etc.
### Step 5: Counting the Triplets
We can systematically check for triplets of primes that satisfy the A.P. condition.
For example:
- Triplet \( (3, 5, 7) \)
- Triplet \( (5, 7, 11) \)
- Triplet \( (11, 13, 17) \)
- Triplet \( (101, 103, 107) \)
- Triplet \( (103, 107, 109) \)
Continuing this process, we can find all valid triplets.
### Conclusion
After analyzing the primes and counting the valid triplets, we find that there are **five** distinct triplets of prime numbers that can be formed such that they are in A.P.
### Final Answer
The number of triplets of prime numbers that can be formed such that the terms of the triplet are in A.P. is **5**.
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