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.

Alternately, 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!

### Prime Gap

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

Prime gapn = prime numbern+1 – prime numbern

### 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 = π ( 1 – ( 1 / (p -1)2 )

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 \[\frac{x}{(ln x)^{2}}\] ～2C₂\[\int_{2}^{x}\] \[\frac{dt}{(ln t)^{2}}\]

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

As the 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 = 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 = (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 = 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 = 144 – 2 / 2

p = 71

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

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1. Which is the Largest Twin Prime Number?

Ans. Two distributed computing projects – Prime Grid and Prime Search, started in 2007 had come up with some large twin prime numbers. Since 2018, 2996863034895 · 21290000 ± 1 is the highest twin prime pair which has 388,342 decimal numbers. Moreover, 808,675,888,577,436 pairs of twin prime numbers occur below 10¹⁸.

2. What Do You Mean By an Isolated Prime Number?

Ans. If neither prime – 2, and prime + 2 gives out a prime number, then the same is called as an isolated prime number or single prime number or a non-twin prime number. This number is not a part of a twin prime pair.

Some of the known isolated prime numbers are 2, 23, 37, 47, 53, 67, 79, 83, 89, 97, …n.

3. What are Coprime and Twin Prime Numbers?

Ans. Coprime numbers are those numbers who share a common factor and the same is only the integer 1. For instance, 21 and 22 are coprime numbers.

With regards to twin primes, they are a pair of numbers having a difference of 2 between them. For instance, 3 and 5 are twin primes.