What do Mersenne primes have to do with perfect numbers?

What do Mersenne primes have to do with perfect numbers?

Lesson Summary If a prime number can be written as 2n – 1 for some n, the prime number is a Mersenne prime. If the sum of divisors of a number (excluding the number itself) equals the number, the number is a perfect number. Perfect numbers are related to Mersenne primes.

How do you check if a number is a Mersenne prime?

Mersenne Prime is a prime number that is one less than a power of two. In other words, any prime is Mersenne Prime if it is of the form 2k-1 where k is an integer greater than or equal to 2. First few Mersenne Primes are 3, 7, 31 and 127.

Why are 1 and 2 not a prime number?

Originally Answered: Why is the number 1 not a prime number? Because prime numbers, by definition, have two distinct factors, one and itself. Since one does not have two distinct factors, it does not meet the first criteria of the definition even though it satisfies the second.

Why are Mersenne primes important?

Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne primes. Many of the largest known primes are Mersenne primes because Mersenne numbers are easier to check for primality.

What is not a Mersenne prime?

cannot be prime. The first four Mersenne primes are M2 = 3, M3 = 7, M5 = 31 and M7 = 127 and because the first Mersenne prime starts at M2, all Mersenne primes are congruent to 3 (mod 4). Other than M0 = 0 and M1 = 1, all other Mersenne numbers are also congruent to 3 (mod 4).

How long is the 35th Mersenne prime?

420,921 digits
The new prime number, 21,398,269-1 is the 35th known Mersenne prime. This prime number is 420,921 digits long. If printed, this prime would fill a 225-page paperback book. It took Joel 88 hours on a 90 MHz Pentium PC to prove this number prime.

Why is 10 a deficient number?

In order for a number to be a deficient number, the sum of the proper factors of the number must be smaller than the number, not greater, or equal to the number. The first 20 deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, and 25.

What is a Mersenne prime number?

In mathematics, a Mersenne prime is a prime that is one less than a power of two. For example, 3 = 4 − 1 = 2 2 -1 is a Mersenne prime; so is 7 = 8 − 1 = 2 3 -1. On the other hand, 15 = 16 − 1 = 2 4 -1, for example, is not a prime, because 15 is divisible by 3 and 5.

What is the smallest Mersenne number with a prime exponent?

More generally, numbers of the form Mn = 2n − 1 without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n be prime. The smallest composite Mersenne number with prime exponent n is 211 − 1 = 2047 = 23 × 89 .

How many Mersenne numbers are congruent to 3?

Other than M0 = 0 and M1 = 1, all other Mersenne numbers are also congruent to 3 (mod 4). Consequently, in the prime factorization of a Mersenne number ( ≥ M2 ) there must be at least one prime factor congruent to 3 (mod 4).

How many Mersenne numbers are there in the world?

Named after Marin Mersenne Subsequence of Mersenne numbers First terms 3, 7, 31, 127, 8191 Largest known term 282,589,933 − 1 (December 7, 2018) OEIS index A000668 Mersenne primes (of form 2^p – 1