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Numbers in the form of 2P – 1 where ‘P’ is a natural number, are called Mersenne numbers. How many Mersenne numbers less than 5000 are prime?

Numbers in the form of 2P – 1 where ‘P’ is a natural number, are called Mersenne numbers. How many Mersenne numbers less than 5000 are prime? Correct Answer 4

Calculation:

⇒ When P = 1 ; 2P – 1 = 21 – 1 = 1

⇒ When P = 2 ; 2P – 1 = 22 – 1 = 3

⇒ When P = 3 ; 2P – 1 = 23 – 1 = 7

⇒ When P = 4 ; 2P – 1 = 24 – 1 = 15

⇒ When P = 5 ; 2P – 1 = 25 – 1 = 31

⇒ When P = 6 ; 2P – 1 = 26 – 1 = 63

⇒ When P = 7 ; 2P – 1 = 27 – 1 = 127 

⇒ When P = 8 ; 2P – 1 = 28 – 1 = 255

⇒ When P = 9 ; 2P – 1 = 29 – 1 = 511

⇒ When P = 10 ; 2P – 1 = 210 – 1 = 1023

⇒ When P = 11 ; 2P – 1 = 211 – 1 = 2047

⇒ When P = 12 ; 2P – 1 = 212 – 1 = 4095

⇒ When P = 13 ; 2P – 1 = 213 – 1 = 8191

⇒ Mersenne numbers (M) will be prime for p = 2, 3, 5, and 7.

⇒ For p = 13 onwards, the value of M will cross 5000.

Additional Information

⇒ A prime number is a positive integer having exactly two factors. If p is a prime, then it’s only factors are necessarily 1 and p itself. Any number which does not follow this is termed as composite numbers, which means that they can be factored into other positive integers.

⇒ The primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29....

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