Twin Prime Numbers

  • In mathematics, there are many types of numbers which are classified according to their some special properties only. Prime numbers are the numbers which are having common factors as 1 and the number itself.

For example: 2, 3, 5, 7, 11… and so on these are the prime numbers as they are having factors 1 and the number itself only. Except 2 all prime numbers are odd prime number.

 

  • Composite numbers are those numbers which are having factors or divisors other than 1 and the number itself also.

For example: 4, 6, 8, 9, 10…and so on are the composite numbers as they are having factors other than 1 and the number itself.

 

  • Co-prime numbers are the numbers which may be prime or composite having GCF as 1 only.

For example: (3, 5), (15, 8) are the co-prime numbers as their GCF is 1 only.

Twin Prime:

Twin prime numbers are the prime numbers only which are having the difference between them as 2.

For example:

  • (3, 5) is the twin prime pair as 5-3 = 2, where 3 and 5 are prime numbers.
  • Again, (5, 7) is also the twin prime pair as 7 – 5 = 2, where 5 and 7 are prime numbers.
  • But, (2, 3) is not the twin prime pair as 3 – 2 = 1, where 2 and 3 are prime numbers.
  • Also, (7, 11) is not the twin prime pair as 11 – 7 = 4, although 7 and 11 are prime numbers.

Thus, in general twin prime are written in the form (6n-1. 6n+1) except the pair (3, 5).

For example:

  • Since, (5, 7) is the twin prime pair which can be written as (5, 7) = (6*1–1, 6*1+1)

= (6-1, 6+1)

= (5, 7)

 

  • Also, (11, 13) is the twin prime pair which can be written as (11, 13) = (6*2-1, 6*2+1)

= (12-1, 12+1).

= (11, 13)

 

  • Again, (17, 19) is the twin prime pair which can be written as (17, 19) = (6*3-1, 6*3+1)

= (18-1, 18+1)

= (17, 19)

There is always one composite number is present in between the numbers forming twin prime pair.

For example:

  • (3, 5) is the twin prime pair, in which there is a composite number 4 in between 3 and 5.
  • Also, (5, 7) is the twin prime pair, in which there is a composite number 6 in between 5 and 7.
  • Again, (11, 13) is the twin prime pair, in which there is a composite number 12 between 11 and 13.

Note: (2, 3) is not the twin prime pair as 3-1≠ 2 and also there is no composite number in between 2 and 3. Although 2 and 3 are prime numbers.

 

The sum of prime numbers occurring between the twin prime pair must be divisible by 12 except in the twin prime pair (3, 5).

For example:

  • (5, 7) is the twin prime pair.

Sum = 5 + 7 = 12, and the sum 12 is divisible by 12.

 

  • (11, 13) is the twin prime pair.

Sum = 11 + 13 = 24, and the sum 24 is divisible by 12.

 

  • (17, 19) is the twin prime pair.

Sum = 17 + 19 = 36, and the sum 36 is divisible by 12.


Updated: July 14, 2021 — 10:00 pm

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