What are Twin Primes?

Twin primes are a pair of prime numbers that have a difference of 2 between them. For example, 3 and 5, 41 and 43 are two common pairs of twin prime numbers.

Alternatively, you can also define twin prime numbers having a prime gap of 2. Now, what is meant by the prime gap?

Let’s study it in detail!

Twin prime numbers form the basis of mathematics, this concept is taught very early in schools and students should get an in-depth knowledge of twin prime numbers as these can help students critically understand the language of numbers and can help them get a good score in the examination.

Twin prime numbers can be defined as a set of two numbers that have only one composite number between them. Another definition of twin prime numbers is – the pair of numbers with a difference of two are also called twin prime numbers. The term twin prime was coined by Stackle in 1916. If we put it in a simple manner, twin prime numbers are numbers where two numbers have a difference of two.

The first few twin primes are n+/-1 for n=4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, etc. Explicitly, these are (3, 5), (5, 7), (11, 13), (17, 19) etc.

Properties of Twin Primes

• We have already studied that twin prime numbers are the pairs of prime numbers that have a difference of two. Here are some of the properties of twin primes-

• 5 is the only prime digit which has positive and negative prime differences of 2. Hence, 5 has two prime pairs - (3,5) and (5,7).

• Both 2 and 3 are not a pair of twin prime numbers as there is no composite number between them.

• The basic form in which prime numbers are represented is {6n-1, 6n+1}, 3 and 5 being an exception.

• Prime pairs when added, their sum is always divisible by 12, 3 and 5 being an exception.

Twin Prime Conjecture

Twin prime conjecture is also called Polignac’s conjecture in terms of number theory. The definition of twin prime conjecture states that there are infinite twin prime pairs with a difference of two. The conjecture states that a positive even number m, has infinite pairs of two consecutive prime numbers with difference n. The twin prime conjecture basically says that infinite twin primes exist.

This means as the numbers get larger, primes start to get less frequent and therefore twin primes get rarer. Polignac’s conjecture is named after Alphonse de Polignac in 1849 as he introduced it. Alphonse discovered that any even number can be expressed in infinite ways as the difference between two consecutive primes. It is also known as Euclid’s twin prime conjecture.

Concepts related to twin primes that will help to get a better understanding of the concept are as follows-

Properties of Twin Primes

• Brun's theorem

• Twin prime conjecture

• Other theorems weaker than the twin prime conjecture

• Conjectures

• First Hardy–Littlewood conjecture

• Polignac's conjecture

Prime Gap

The prime gap is nothing but the difference between two consecutive prime numbers. In mathematical form, it can be expressed as:

Prime gap = prime number + 1 – prime number

What are Twin Primes Properties?

The only prime digit having both positive and negative prime differences of two is 5. Therefore, it occurs in two prime pairs – (3, 5) and (5, 7).

As no composite number exists between 2 and 3, these two successive digits are not a pair of twin prime numbers.

All other pairs of twin prime numbers stay in the form of {6n – 1, 6n + 1}, except the pair (3, 5).

If you add two numbers of a prime pair, the result will be divisible by 12. Again, for this case, the pair (3, 5) is an exception.

What is Twin Prime Number Conjecture?

Alphonse de Polignac, a French mathematician, introduced the first statement of twin prime conjecture in 1846. He stated that any even numeral could be illustrated in an infinite number of ways. An example of the same is the difference between two prime numbers (13 – 11 = 5 – 3 = 2).

This theory is sometimes referred to as Euclid’s twin prime conjecture, but it was proved that an infinite number of primes might exist. However, this is not the case for twin prime numbers.

First Hardy-Littlewood Conjecture

This conjecture is named after two English mathematicians, namely G. H Hardy and John Littlewood. Involved in prime constellation distribution, which includes twin primes, this conjecture generalises the twin conjecture.

Consider π2(x) to be the number of prime digits provided p is lesser than equals to x, such that p + 2 also gives a prime number. Therefore, the constant of twin prime C2 can be represented as the following -

C2 = \[\pi \left ( 1-\left ( \frac{1}{\left ( p-1 \right )^{2}} \right ) \right )\]

Here, p is a prime number and greater than equals to 3.

The approximate answer is 0.660161815846869573927812110014…. 

Furthermore, the unique of the initial Hardy-Littlewood conjecture can be illustrated as:

π₂(x)～2C₂ x(lnx)2

 ～2C₂∫x2

 dt(lnt)2

Note: The integer ‘2’ is the one and only even prime number.

As Polignac’s conjecture stated that there are many twin prime pairs with a difference of 2, but Yitang Zhang proved that there is an infinite number of prime pairs, which holds a gap of not less than 70 million.

Various other Prime Types

Cousin Prime

When the difference between two prime numbers is 4, they are termed as cousin primes.

Prime Triplet

The set of prime triplets contains three numbers such that the largest and smallest number has a difference of 6. Two exceptions are (2, 3, 5) and (3, 5, 7).

Solved Numerical on Twin Primes

Problem 1: What are Twin Primes Between 1 and 100?

Solution. The twin prime pairs between 1 and 100 are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61) and (71, 73).

Do it Yourself

Problem 2: Find out whether the following numbers are the addition of twin prime numbers. (a) 36 (b) 120 (c) 84 (d) 144

Solution. You can use the formula p + (p + 2) to find out whether the above-mentioned numbers are twin primes. Take a look!

(a) 36 

p + (p + 2) = 36

2p + 2 = 36

2p = 36 – 2

P = \[\mathrm{\frac{34}{2}}\]

P = 17

So, substituting the value in (p + 2) we get 17 + 2 = 19.

(b) 120

p + (p + 2)

2p + 2 = 120

P = \[\mathrm{\frac{120-2}{2}}\]

P = 59

Again, substituting the value in (p + 2) we get 59 + 2 = 61.

(c) 84

p + (p + 2) = 84

2p + 2 = 84

p = \[\mathrm{\frac{84-2}{2}}\]

p = 41

Putting the value of p in p + 2 we get 41 + 2 = 43

(d) 144

p + (p + 2) = 144

2p + 2 = 144

p = \[\mathrm{\frac{144-2}{2}}\]

p = 71

Putting the value of p in p + 2, we get 71 + 2 = 73

Conclusion:

By going through the information provided above, you must have been able to understand what twin primes are. You can also refer to solved questions for practice.