What is the smallest positive common difference of a 6-term arithmetic progression consisting entirely of (positive) prime numbers?
To find the smallest positive common difference of a 6-term arithmetic progression consisting entirely of positive prime numbers, we need to consider some properties of prime numbers.
First, let's understand what an arithmetic progression is. An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. For example, 3, 7, 11, 15 is an arithmetic progression with a common difference of 4.
In this case, we are looking for an arithmetic progression consisting entirely of prime numbers. Prime numbers are natural numbers greater than 1 that have no divisors other than 1 and themselves. For example, 2, 3, 5, 7 are prime numbers.
Since we are looking for the smallest positive common difference, we should start by considering small prime numbers and their differences. Let's list the prime numbers from smallest to largest: 2, 3, 5, 7, 11, 13, ...
Now, let's consider possible differences between consecutive prime numbers:
- The difference between 2 and 3 is 1, which is not a prime number.
- The difference between 3 and 5 is 2, which is a prime number.
- The difference between 5 and 7 is 2, which is a prime number.
- The difference between 7 and 11 is 4, which is not a prime number.
- The difference between 11 and 13 is 2, which is a prime number.
From this observation, we can see that the smallest positive common difference for a 6-term arithmetic progression consisting entirely of prime numbers is 2. The prime numbers in this progression would be: 3, 5, 7, 11, 13, 17.
To further confirm, we can continue checking larger prime numbers, but the pattern of 2 as the smallest positive common difference will hold true.
Therefore, the answer is 2.