Twin Prime Numbers
Twin Prime Numbers are the set of two numbers that have exactly one composite number between them. They can also be defined as the pair of numbers with a difference of 2. The name twin Prime was coined by Stackel in 1916. In simple words, we can say that where two numbers have a difference of 2, they are said to be Twin Primes. The word twin prime is also used to describe one of the twin primes i.e. twin prime is a prime with a prime gap of 2.
First Pairs of Twin Prime Numbers
The first twin Prime Numbers are {3,5}, {5,7}, {11,13}, and {17,19}. It has been conjured that there are infinite twin primes. According to sieve techniques, the sum of the reciprocals of twin primes converges hence all the pairs of twin primes are in the form of {6n-1, 6n+1} except the first pair of twin prime which is (3, 5). It has been conjured in the famous conjecture of twin primes that there are infinite twin primes. Further using sieve techniques it has been proved that the sum of the reciprocals of twin primes converges. Other than the first pairs, all pairs of twin primes have the form {6n-1, 6n+1}.
Properties of Twin-Prime Numbers
We know that twin primes are pairs of prime numbers with a difference of two. There are few basic properties for twin primes. Let us discuss the properties of twin primes in detail.
- 5 is the only prime number that has a positive as well as a negative prime gap of two and hence is the only prime to occur in two twin primes pair.
- Every twin primes pair other than {3, 5} is of the form {6n-1, 6n+1}
- Pair of numbers is not considered a twin prime, if there is no composite number between them, for example, {2, 3} can not be considered a twin prime pair. as there is no composite number between them.
- The sum of each twin prime pair except {3,5} is divisible by 12 as (6n-1) + (6n+1) = 12n.
What is Twin Prime Number Conjecture?
Twin prime conjecture, is another word for Polignac’s conjecture, in number theory. According to the twin primes definition, there are infinite twin prime pairs with a difference of 2. Polignac's conjecture was introduced by Alphonse de Polignac in 1849. The conjecture states that, for positive even number m, there are infinitely many pairs of two consecutive prime numbers with difference n. Twin prime conjecture, also known as Polignac’s conjecture asserts that there are infinitely many twin primes. Now, 3 and 5, 5 and 7, 11 and 13, and 17 and 19 are all twin primes. As numbers get larger, primes become less frequent and thus. twin primes get rarer.
The first statement of the twin prime conjecture was given in 1846 by French mathematician Alphonse de Polignac, who wrote that any even number can be expressed in infinite ways as the difference between two consecutive primes. When the even number is 2, this is the twin prime conjecture; that is, 2 = 5 − 3 = 7 − 5 = 13 − 11 = and so on. Although the conjecture is sometimes called Euclid’s twin prime conjecture, he gave the oldest known proof that there exists an infinite number of primes but did not conjecture that there is an infinite number of twin primes.
Difference Between Twin Prime Numbers and Co-Prime Numbers
Twin prime numbers are the pair of prime numbers with a difference of 2, whereas co-prime numbers are the numbers having only 1 as a common factor. All twin primes are co-primes numbers but all co-primes are not twin primes. Co-primes may not be prime numbers, they have the GCD=1. All twin primes are co-primes numbers but vice versa is not true. Co-primes need not be prime numbers, they can be any numbers with their GCD=1.
For example, 13 and 14 are two co-prime numbers. The only common factor between the two numbers is 1 and hence they are co-prime. Here, (13,14) are not twin primes.
Related Articles on Twin Prime Numbers
Solved Examples on Twin Prime Numbers
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Example 2. With the help of the given prime numbers chart, prepare the list of twin prime numbers from 1-100 that gives the difference of 2.
Solution:
To find the first 8 pairs of twin prime numbers we need to check the difference between the two prime numbers is 2. Let us first take 3 and 5. 3 and 5 both are prime numbers and the difference is 2. Similarly, we will try with all the primes numbers listed in the given chart. Therefore, the list of twin primes from 1-100 is (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), and (71, 73).
FAQ's on Twin Prime Numbers
What are Twin Prime Numbers?
Pair of prime numbers with a difference of two.
What are the Examples of Prime Number Pairs That Have a Difference of Two?
Examples of such pairs which are also called twin prime numbers are {3,5}, {5,7}, {11,13}, {17,19} and so on.
What is the Difference Between Co-Prime and Twin Prime Numbers?
All twin prime numbers are co-prime numbers but all co-prime numbers are not twin prime numbers. Co-primes may not be prime numbers, they have the GCD=1.
Can you List the Twin Prime Numbers From 1-100?
The list of twin prime numbers from 1-100 is (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), and (71, 73).
How Do You Find Twin Prime Numbers?
To find the twin prime numbers we need to check the numbers are prime numbers and the difference between the two prime numbers is 2. Let us first take 3 and 5. 3 and 5 both are prime numbers and the difference is 2.
Is 71 and 73 a Twin Prime Number Pair?
Yes, 71 and 73 form a twin prime number pair. 71 is a prime number and 73 is also a prime number. A pair of prime numbers with a difference of two are twin primes. Hence, 71 and 73 are twin primes.