TWIN PRIME NUMBERS

There are six pairs of twin prime numbers from 1 to 50. They are

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

There are two pairs of twin prime numbers from 51 to 100. They are

(59, 61), (71, 73)

There are seven pairs of twin prime numbers from 101 to 200. They are

(101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193), (197, 199)

There are four pairs of twin prime numbers from 201 to 300. They are

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

There are two pairs of twin prime numbers from 301 to 400. They are

(311, 313), (347, 349)

There are three pairs of twin prime numbers from 401 to 500. They are

(419, 421), (431, 433), (461, 463)

There are eleven pairs of twin prime numbers from 501 to 1000. They are

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

In all, there are 35 pairs of twin prime numbers from 1 to 1000.

Properties of Twin Primes

●  A pair of numbers is not considered to be twin prime, if there is no composite number between them.

For example, (2, 3) can not be considered as twin prime pair. Because, there is no composite number between them. Moreover, the difference between 2 and 3 is not equal to 2.

● 5 is the only prime number that is available in two different pairs of twin primes. They are

(3, 5) and (5, 7)

●   Every pair of twin primes other than (3, 5) is in the form of

(6n - 1, 6n + 1),

where n is any natural number.

Note :

Every twin prime will be in the form of (6n - 1, 6n + 1). But, substituting natural numbers for n into (6n - 1, 6n + 1) will always NOT yield a twin prime.

Example :

When n = 4,

(6n -1, 6n + 1) = (23, 25)

Here, (23, 25) is not a twin prime. Because 25 is not a prime number, it is a composite number.

●  The sum of two primes numbers in each pair of twin primes apart from (3, 5) is divisible by 12.

(6n - 1) + (6n + 1) = 6n - 1 + 6n + 1

= 12n

12n is a multiple of 12. So it is divisible by 12.

Difference Between Coprime and Twin Prime Numbers

Twin Prime Numbers :

Twin prime numbers are the two prime numbers which have the difference of 2.

Examples :

(3, 5), (5, 7), (11, 13), (17, 19)

Coprime Numbers :

Co prime numbers are any two numbers which have the greatest common divisor 1.

Examples :

(1, 2), (2, 3), (4, 9), (21, 32)

(2, 4) are not coprime numbers. Because, the greatest common divisor of 2 and 4 is 2. 

Solved Problems

Problem 1 :

Are 17 and 19 twin prime numbers? Explain.

Solution :

We know that the two prime numbers with difference 2 are called twin prime numbers.

Both 17 and 19 are prime numbers.

19 - 17 = 2

The difference between 17 and 19 is 2.

Therefore, 17 and 19 are twin prime numbers.

Problem 2 :

Are 2 and 3 twin prime numbers? Explain.

Solution :

Both 2 and 3 are prime numbers.

3 - 2 = 1 ≠ 2

Even though 2 and 3 are prime numbers, the difference between them is not 2.

So, the prime numbers 2 and 3 are NOT twin prime numbers.

Problem 3 :

Are 23 and 29 twin prime numbers? Explain.

Solution :

Both 23 and 29 are prime numbers.

29 - 23 = 6 ≠ 2

Even though 23 and 29 are prime numbers, the difference between them is not 2,

So, the prime numbers 23 and 29 are NOT twin prime numbers.

Problem 4 :

By what number is the sum of two primes numbers in each pair of twin primes apart from (3, 5) divisible? Give examples.

Solution :

The sum of two primes numbers in each pair of twin primes apart from (3, 5) divisible by 12.

Examples :

(5, 7) ----> 5 + 7 = 12 (divisible by 12)

(17, 19) ----> 17 + 19 = 36 (divisible by 12)

(41, 43) ----> 41 + 43 = 84 (divisible by 12)

Problem 5 :

For any natural number k, is (6k - 1) a prime number? If so, give some examples.

Solution :

For any natural number k, (6k - 1) is a prime number.

Examples :

k = 1 :

6(1) - 1 = 6 - 1 = 5 (prime number)

k = 2 :

6(2) - 1 = 8 - 1 = 7 (prime number)

k = 3 :

6(3) - 1 = 18 - 1 = 17 (prime number)

Problem 6 :

For any natural number p, is (6p + 1) a prime number? If so, give some examples.

Solution :

For any natural number, (6p + 1) is a prime number.

Examples :

p = 1 :

6(1) + 1 = 6 + 1 = 7 (prime number)

p = 2 :

6(2) + 1 = 12 + 1 = 13 (prime number)

p = 3 :

6(3) + 1 = 18 + 1 = 19 (prime number)

Problem 7 :

For any natural number a, will all the pairs of twin prime numbers in the form (6a - 1, 6a + 1)? If so, give some examples.

Solution :

For any natural number a, all the pairs of twin prime numbers will be in the form (6a - 1, 6a + 1).

Examples :

Take the twin prime (11, 13).

11 is 1 less than 12, which is a multiple of 6

13 is 1 more than 12, which is a multiple of 6

(11, 13) is in the form of (6n - 1, 6n + 1)

Take the twin prime (29, 31).

29 is 1 less than 30, which is a multiple of 6

31 is 1 more than 30, which is a multiple of 6

(29, 31) is in the form of (6n - 1, 6n + 1)

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