Question

Let \(p\) and \(q\) be two prime numbers. If \(p=q+2\) then \(p\) and \(q\) are called "twin prime numbers." Find two pairs of twin prime numbers.

Short answer

The pairs of twin prime numbers are (3, 5) and (5, 7).

Step-by-step solution

Understanding Prime Numbers

Prime numbers are numbers that have only two factors: 1 and the number itself. For example, the first six prime numbers are 2, 3, 5, 7, 11, and 13.

Identifying Twin Primes

Twin primes are pairs of prime numbers that differ by two. This difference means that if you have a prime number and you add 2 to it, if the result is also a prime number, then you have a pair of twin primes. This pair satisfies the given equation \(p=q+2\). Working through the list of given prime numbers, start with 2. Add 2 to 2, you get 4 which is not a prime number. Move on to 3, add 2 to 3 and you get 5 which is a prime number, so 3 and 5 form a pair. The same goes for 5, adding 2 to 5, gives 7 which is also a prime number, hence another pair.

Giving Final Pairs of Twin Primes

The pairs (3, 5) and (5, 7) are solutions to this problem since both these pairs satisfy the definition of twin primes, i.e., one number is 2 greater than the other and both numbers are prime.

Key concepts

Prime Number Definition

Prime numbers are the building blocks of all numbers. They are special because they can only be divided evenly by 1 and themselves. This means that they have exactly two distinct positive divisors.
For example:

• 2 is a prime number because its only factors are 1 and 2.
• 3 is also prime because it can only be divided by 1 and 3 without leaving a remainder.
• Larger numbers like 11 and 13 are prime for the same reason—they have no divisors other than 1 and the number itself.

Prime numbers are fundamental in number theory. They play a crucial role in various fields like cryptography and coding. Identifying and understanding them is essential in math because they are like the "atoms" of mathematics from which other numbers are constructed.

Twin Primes

Twin primes are a fascinating subset of prime numbers. These are pairs of prime numbers that differ by exactly 2. In other words, if you start with a prime number and add 2, the result is also a prime number, then you've got a twin prime pair.

Here are a couple of examples to help clarify:

• Starting with the prime number 3, add 2 to get 5. Both numbers are prime, making (3, 5) a twin prime set.
• Similarly, starting with 5 and adding 2 gives us 7. Again, both are prime, so (5, 7) is another twin prime pair.

Twin primes are rare, and mathematicians are still intrigued by them. They seek to understand how many such pairs exist. It's an unsolved question if there are infinitely many twin primes, but they keep us curious and engaged.

Algorithm Solving

To find twin primes, an algorithmic approach is very useful. Algorithms are step-by-step procedures for calculations. When solving for twin primes, an algorithm simplifies the process by setting a clear path.

Here’s a basic approach to finding twin primes:

• First, create a list of prime numbers. You can use methods like the Sieve of Eratosthenes to generate primes efficiently.
• Then, for each prime number in the list, check if adding 2 results in another prime number.
• If it does, you’ve found a pair of twin primes.

This systematic approach saves time and minimizes errors. Algorithms make complex computations simple, and in this case, help us uncover the delightful pattern of twin primes without the need for tedious manual calculation.