Understanding Twin Prime Numbers

The question asks us to identify a pair of twin prime numbers from the given options. To answer this, we first need to understand what prime numbers are and what twin prime numbers are.

What is a Prime Number?

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples include 2, 3, 5, 7, 11, and so on.

What are Twin Prime Numbers?

Twin prime numbers are a pair of prime numbers that differ by 2. In other words, a pair of prime numbers (p, p+2) are twin primes if both p and p+2 are prime numbers. The smallest twin prime pair is (3, 5).

Analyzing the Options

Let's examine each given option to see if it fits the definition of twin prime numbers:

• Option A: (4, 9)

  Here, 4 is divisible by 2 and 4, so it is not a prime number. 9 is divisible by 3 and 9, so it is also not a prime number. Since neither number is prime, this is not a pair of twin prime numbers.

• Option B: (2, 3)

  Here, 2 is a prime number (only divisible by 1 and 2). 3 is also a prime number (only divisible by 1 and 3). Both are prime. Let's find the difference: \(3 - 2 = 1\). The difference is 1, not 2. Therefore, this is not a pair of twin prime numbers.

• Option C: (4, 6)

  Here, 4 is not a prime number (divisible by 2 and 4). 6 is not a prime number (divisible by 2, 3, and 6). Since neither number is prime, this is not a pair of twin prime numbers.

• Option D: (3, 5)

  Here, 3 is a prime number (only divisible by 1 and 3). 5 is a prime number (only divisible by 1 and 5). Both are prime numbers. Let's find the difference: \(5 - 3 = 2\). The difference is 2. Since both are prime and their difference is 2, this is a pair of twin prime numbers.

Conclusion

Based on our analysis, the pair (3, 5) consists of two prime numbers whose difference is exactly 2. Thus, (3, 5) is a pair of twin prime numbers.

Revision Table: Twin Primes Concepts

Additional Information: More About Twin Primes

The question of whether there are infinitely many twin prime numbers is a famous unsolved problem in mathematics known as the Twin Prime Conjecture. Most mathematicians believe the conjecture is true. While it hasn't been proven that there are infinitely many twin primes, significant progress has been made, such as showing there are infinitely many pairs of primes that differ by less than a certain small number.

Note that 2 is the only even prime number. All other prime numbers are odd. Therefore, twin prime pairs (except for (3, 5)) will always consist of two odd numbers, and their difference is 2.

Twin prime pairs follow a pattern for pairs greater than (3, 5): they are of the form \((6n - 1, 6n + 1)\) for some natural number n. For example, for n=1, \((6(1) - 1, 6(1) + 1) = (5, 7)\). For n=2, \((6(2) - 1, 6(2) + 1) = (11, 13)\).