Prime Numbers

Hi friends, meeting you here after a long time, but assure you that will meet you here regularly (almost every month) with new interesting topics in mathematics.

Last week only a famous American mathematician died in a road accident. His name was John Nash. He was a Noble laureate. He won noble prize in economics, in which he was never officially trained!!! John Nash

We are going to discuss today prime numbers. I think numbers are very interesting to play with, you may like this topic. From upper primary years we start understanding types of numbers. But important aspect of classifying numbers we get by divisibility.  We know based on this there are two types of numbers, Prime umbers and Composite numbers.

Prime Numbers:

A number which can be divided evenly only by 1 or itself and must be a whole number greater than 1.

Composite Numbers:

It is a number when it can be divided evenly by numbers other than 1 and itself.

  • Any whole number, greater than one is either Prime or Composite number.
  • 1 is neither Prime nor Composite number.
  • Did you know there are 70 different types of Prime numbers?

Some of them are as follows

Balanced Prime, Chen Prime, Circular Prime, Cousin Prime, Co-prime, Cuban prime, Cullen Prime. Dihedral Prime, Eisenstein Prime, Factorial prime, Fibonacci Prime, Fortunate Prime, Gaussian prime, Good Prime, Happy Prime, Higgs Prime, Mersenne Prime, Minimal Prime, Palindromic  Prime,   Ramanujan Prime, Strong Prime , Super Prime, Twin Prime, Unique Prime, Wilson Prime, and so on.

We know two types of primes, co-primes and twin primes. 

Co-primes: In number theory two integers a and b are said to be relatively prime, mutually prime or co-prime if the only positive integer that evenly divides both of them is 1. That is only positive common factor of two numbers is 1.

            For example 14 and 15, but 14 and 21 are not. The number 1 and -1 co-prime to every integer and they are only integers to be co-prime with 0.

  • All prime numbers are co-prime to each other.
  • Any two consecutive integers are always co-prime numbers.
  • Sum of any two co-prime numbers is always co-prime with their product.
  • 1 is co-prime with all integers.
  • a and b (natural numbers) are co-prime if 2a-1 and 2b-1 are co-primes.

Cousin Primes:

In mathematics Cousin Primes are prime numbers that differ by four. Compare these with twin primes, pairs of primes differ by two. The pairs of prime numbers that differ by six are called sexy primes.

Cousin Primes (3,7);(7,11); 13,17); (19,23); (43,47); 67,71); (79,83); (97,101); …………….(907,911); (937,941); (967,971). These are up to 1000. Try and find other numbers in between. The only prime number belongs to two cousin prime pairs is ‘7’.

As of May 2009 largest known Cousin prime was (p,p+4) form

i) 474435381.298394 -1

ii) 47445381. 298394- 5

These numbers have 29629 digits and were found by Angle, Jobling and Augustine.

Mersenne Primes:

In mathematics Mersenne Prime is a number of the form Mn= 2n-1. If n is prime number then so is the 2n-1. Mersenne primes are sometimes have additional requirement that

Additional requirement that n be prime, equivalent they be Pernicious Mersenne numbers.

Smallest Composite Pernicious Mersenne number is

211-1=2047=23x89 is a pseudo prime number.

As of October 2014, 48 Mersenne Primes are known

  • The ten largest known prime numbers

Rank

Prime number

Found by

Found date

Number of digits

Reference

1st

257,885,161 − 1

GIMPS

2013 January 25

17,425,170

[2]

2nd

243,112,609 − 1

GIMPS

2008 August 23

12,978,189

[2]

3rd

242,643,801 − 1

GIMPS

2009 April 12

12,837,064

[6]

4th

237,156,667 − 1

GIMPS

2008 September 6

11,185,272

[6]

5th

232,582,657 − 1

GIMPS

2006 September 4

9,808,358

[6]

6th

230,402,457 − 1

GIMPS

2005 December 15

9,152,052

[7]

7th

225,964,951 − 1

GIMPS

2005 February 18

7,816,230

[7]

8th

224,036,583 − 1

GIMPS

2004 May 15

7,235,733

[7]

9th

220,996,011 − 1

GIMPS

2003 November 17

6,320,430

[7]

10th

213,466,917 − 1

GIMPS

2001 November 14

4,053,946

[7]

 

Twin Primes :

A twin prime is a prime numberthat has a prime gapof two, in other words, differs from another prime number by two, for example the twin prime pair (41, 43). Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes appear despite the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger due to the prime number theorem.

The first few twin prime pairs are:

(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), …  A077800.

An isolated prime is a prime number p such that neither p − 2 nor p + 2 is prime. In other words, p is not part of a twin prime pair. For example, 23 are an isolated primes since 21 and 25 are both composite.

The first few isolated primes are

2, 23, 37, 47, 53, 67, 79, 83, 89, 97, …  A007510