Last updated on July 4th, 2025
A twin prime pair contains two prime numbers with a composite number in between. The difference between a pair of twin prime numbers is 2. In 1916, Stackel introduced the term to mathematics, which refers to a set of prime numbers that differ by 2. A twin prime pair is a pair of numbers with a prime gap of 2. In this topic, we will learn about twin prime numbers and their properties in detail.
A twin prime pair contains two prime numbers with a difference of 2. (3, 5), (17, 19) are some examples of twin prime pairs. The first few twin prime pairs are (3, 5), (5, 7), (11, 13), and (17, 19). Except for the first pair (3, 5), the pattern (6n - 1, 6n + 1) is followed by all twin prime pairs.
A twin prime pair contains two prime numbers, while a prime triplet has three prime numbers, and each consecutive prime number has a difference of 2, such as (3, 5, 7).
Prime pairs with a difference of 4 are known as cousin primes.
Examples of cousin primes are (3, 7), (7, 11), (13, 17), and so on.
The major differences between twin prime and co-prime numbers are tabled below:
Characteristics | Twin prime | Co-prime numbers |
Definition | Twin prime numbers are prime numbers that differ by 2. | Co-prime numbers are numbers that only have one common factor, that is 1. |
Relationship | They are a subset of co-prime numbers. | The co-prime numbers can be either prime or composite. |
Feature | Both numbers in a twin prime pair are prime. | The co-prime numbers can be either prime or composite. |
Greatest common divisor (GCD) | The GCD of twin prime pairs will be 1. | The GCD of co-prime numbers will always be 1. |
Property | All twin primes are co-primes. | Co-prime numbers are not always twin primes. |
Examples | (3, 5), (5, 7), (11, 13), (17, 19) | (6, 25), (7, 11), (13, 14), (15, 16) |
Twin prime number pairs are a set of two prime numbers with a difference of 2 between them. The key properties of twin prime numbers are:
A twin prime pair contains two prime numbers with a difference of 2 between each prime. We can check if a given pair of numbers is a twin prime pair by following these steps:
Step 1: Check whether the numbers in a twin prime pair are prime or not. A prime number is a number that only has two factors: 1 and the number itself.
For example, in the pair (11,13),
First, check if 11 and 13 are prime or not.
11 is a prime number; it is only divisible by 1 and 11.
Similarly, 13 is divisible only by 1 and 13. Therefore, 13 is also a prime number.
Step 2: Ensure that the difference between 11 and 13 is 2.
13 - 11 = 2
Step 3: Write the result as (11, 13) is a twin prime pair.
In 1849, Alphonse de Polignac introduced the twin prime conjecture, also known as Polignac’s conjecture. In number theory, there are an infinite number of twin prime pairs that have a difference of 2 with each prime. According to Polignac’s conjecture, for any positive even number ‘m’, there are infinite pairs of primes with a difference of ‘m’. This conjecture states that there are infinitely many twin primes. The occurrence of twin primes and prime pairs becomes less common as numbers get bigger.
Alphonse de Polignac said that the difference between two consecutive primes can be used to express any even number in infinite ways. If the even number is 2, the twin prime conjecture applies,
2 = 5 - 3 = 7 - 5 = 13 - 11 = and so on.
While Euclid’s twin prime conjecture established that there are an infinite number of primes. Unfortunately, he did not give any proof that there is an infinite number of twin primes.
Twin primes are an important concept in mathematics that is closely related to the study of prime numbers. They have a wide variety of real-world applications. These are:
By learning the properties of twine prime numbers, students can easily distinguish them from prime and composite numbers. Sometimes, students mistakenly identify non-twin primes as twin primes. Here are some common errors and helpful solutions to avoid these mistakes:
Identify all twin prime pairs in the given set of numbers: (11, 13, 17, 19, 29, 31, 41, 43)
(11,13), (17, 19), (29, 31), (41, 43)
If the numbers have a difference of 2, then it is a twin prime pair.
Now we can check the difference:
(11, 13) = difference is 2.
(17, 19) = difference is 2.
(29, 31) = difference is 2.
(41, 43) = difference is 2.
Thus, the twin prime pairs in the given set are:
(11,13), (17, 19), (29, 31), and (41, 43)
Find the next twin prime pair after (29, 31).
(41, 43)
A twin prime pair has two prime numbers that have a difference of 2.
Here, we have to find the next twin prime pair after (29, 31).
Now, we need to find the next prime number after 31.
As we know, 37 is the next prime number.
37 + 2 = 39
Let us check whether 39 is a prime number or not.
39 is divisible by 3 and 13, so it is not a prime number.
Therefore, (37, 39) is not a twin prime pair.
Now, move on to the next prime number after 37.
41 is a prime number, and the next prime number after 41 is 43.
Next, check the difference between 41 and 43.
43 - 41 = 2
Hence, the first twin prime pair after (29, 31) is (41, 43).
Find the sum of the first two pairs of twin primes.
20
The first two twin prime pairs are:
(3, 5) and (5, 7)
First, we can add each pair of numbers:
For the pair (3, 5):
3 + 5 = 8
For the pair (5, 7):
5 + 7 = 12
Next, add both the sums together.
8 + 12 = 20
Hence, the sum of the first two pairs of twin primes is 20.
Identify all twin prime pairs in the given set of numbers: (51, 59, 61, 71, 73, 85)
(59, 61) (71, 73) are the twin prime pairs.
First, we can list the prime numbers from the given set of numbers.
51 is not a prime number because it is divisible by 3 and 17.
59 is a prime number, and 1 and 59 are its factors.
61 is a prime number, and the factors are 1 and 61.
71 is a prime number since its factors are 1 and 71.
73 is a prime number, and its factors are 1 and 73.
85 is not prime because it is divisible by 5 and 17.
Hence, the prime numbers in the set are 59, 61, 71, and 73.
Next, we can check for twin prime pairs.
(59, 61) is a twin prime pair.
Now, find the difference:
61 - 59 = 2
73 - 71 = 2
The difference between each twin prime number is 2.
Therefore, the twin prime pairs in the given set are (59, 61) and (71, 73).
Find the product of the first three pairs of twin primes.
75,075
The first three pairs of twin primes are:
(3, 5)
(5, 7)
(11, 13)
Now, we can find the product of each pair.
Product of (3, 5) = 3 × 5 = 15
Product of (5, 7) = 5 × 7 = 35
Product of (11, 13) = 11 × 13 = 143
Next, multiply each product together.
15 × 35 × 143 = 75,075
Hence, the product of the first three twin prime pairs is 75,075.
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.