Twin Prime: List, Properties & Examples

Twin prime numbers are defined as the set of prime numbers that have exactly one composite number in between them. In other words, we can define these numbers as the prime numbers that have a difference of two between them. 

• We can also use the number to describe one of the two twin prime numbers.
• These numbers are either two less or two more than another prime number.
• It is also known as prime twin or prime pair.
• The term was coined by Stackel in 1916.
• They are considered rare when evaluating large-range values.
• It has been determined that there are infinitely many different types of twin primes.
• {3,5}, {5,7}, {11,13} and {17,19} are first set of these numbers.

Key Terms: Prime Numbers, Twin Primes, Prime Triplets, Conjectures, Co-Prime Numbers, Numbers, Prime Gap, First Hardy-Littlewood Conjecture, Twin Prime Number Conjecture, Composite Numbers

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Properties of Twin Prime

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Twin prime are prime numbers that have a difference of two between them. These numbers have something in common between them. Numbers that belong to these categories depict some common properties, which are as follows:

• Every twin prime number except (3,5) can be expressed in one given form, which is 6n+1 and 6n-1.
• The sum of every twin prime number except (3,5) can be expressed in the form of 12n as 6n-1) + (6n+1) = 12n.
• Any two prime numbers that do not have a composite number between them cannot be considered as twin prime.
• 5 is the only prime number that has both positive and negative prime gaps. 
• It has a negative gap of 2 with 3 and a positive gap of 2 with 7.
• So, it is the only prime number that occurs in two prime twins.

Twin Prime 

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First Few Pairs of Twin Prime

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To find the first few pairs of twin primes, we must look at the prime numbers that occur first. The first set of twin prime numbers are (3,5). The next set of twin prime numbers are (5,7), (11,13), (17,19) etc. 

• The next pair of twin prime numbers can be found because they can be expressed in the form of (6n-1, 6n+1). 
• All the twin prime numbers can be expressed in the form of (6n-1, 6n+1) except the pair (3,5). 
• Further, there are infinite numbers of twin prime numbers that can be found by using this conjecture.

As a result first few pairs of twin prime numbers are as follows:

• (3,5), (5,7), (11,13), (17,19), (29,31), (41,43), (59,61), (71,73), (101,103), (107,109), (137,139), (149,151), (179,181), (197,199)…..

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Twin Prime Numbers List

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The twin prime numbers start from (3,5). The list of the twin prime numbers less than 1000 are as follows:

Twin prime numbers from 1 to 50

The number are as follows:

• (3,5), (5,7), (11,13), (17,19), (29,31), (41,43)

Twin prime numbers from 51 to 100

The number are as follows:

• (59,61), (71,73)

Twin prime numbers from 101 to 200

The number are as follows:

• (101,103), (107,109), (137,139), (149,151), (179,181), (197,199)

Twin prime numbers from 201 to 300

The number are as follows:

• (227,229), (239,241), (269,271), (281,283)

Twin prime numbers from 301 to 400

The number are as follows:

• (311,313), (347, 349)

Twin prime numbers from 401 to 500

The number are as follows:

• (419, 421), (431, 433), (461, 463)

Twin prime numbers from 501 to 1000

The number are as follows:

• (521, 523), (569, 571), (599, 601), (617, 619), (641,643), (659, 661), (809, 811), (821,823), (827,829), (857, 859), (881,883).

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Prime Gap

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Prime gap is defined as the difference between two consecutive prime numbers. The smallest and only odd prime number is of gap one. As a result, the required number of prime gap of n numbers can occur at much smaller than n!. 

• Similarly, the prime gap for other successive prime numbers can be found out by finding the difference between them.
• According to the prime number theorem, the average prime gap between a prime and the next prime will be ln(p).
• As per analysis, the prime gap of 60 numbers is as follows:

{1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2}

• It can mathematically be represented as follows:

Prime gap = prime number + 1 – prime number

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Prime Triplets

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Prime triplets are a set of three numbers that can be represented in a common factor. The form that is used in expressing the triplets is (n, n+2, n+6) or (n, n+4, n+6).

• In these numbers, the difference between the smallest and largest prime number is six.
• Twin prime, sexy prime and cousin prime are three types of prime triplets.
• One of the numbers in the group is a multiple of three.
• (2, 3, 5) and (3, 5, 7) are two sets of prime triplets.

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Conjectures

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Conjectures are divided into two categories which are as follows:

Twin Prime Number Conjecture

Twin prime number conjecture is also known as Polignac’s conjecture in number theory. According to this conjecture, for each positive even number m, there are infinitely many pairs of two consecutive prime numbers with difference n.

• According to this conjecture, there are infinitely many twin primes.
• As the numbers go on increasing, the occurrence of twin prime numbers decreases.
• The term was coined by French mathematician Alphonse de Polignac in 1896.
• He expressed that even can be explained by the difference between two consecutive prime numbers.

First Hardy-Littlewood Conjecture

The First Hardy-Littlewood conjecture is an extension of the twin prime number conjecture. It is related to the distribution of prime numbers, including twin primes with respect to the prime number theorem.

• Let p(x) denote the number of primes p<=x, such that p+2 is also prime (twin prime).
• Then the twin prime constant c, is given by C= summation p>=3 (1-1/(p-1)^2.

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Difference between Twin Prime and Co-Prime Numbers

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The difference between Twin Prime and Co-Prime Numbers are as follows:

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Things to Remember

• Twin prime numbers are the prime numbers that have a composite number between them, and the difference between them is 2.
• These numbers can be expressed in the form of (6n-1, 6n+1) except (3,5).
• 5 is the only number that has both positive and negative prime gap 2.
• For every positive prime number m, there are an infinite number of twin prime numbers with a difference of n. 
• As the numbers increase, the occurrence of twin prime decreases.
• Every twin prime number is co-prime,coprimes but not every coprime is a twin prime.