It is currently an open problem as to whether there are an infinite number of Mersenne primes and even perfect numbers. For instance, in the case of p = 2, 2 2 − 1 = 3 is prime, and 2 2 − 1 × (2 2 − 1) = 2 × 3 = 6 is perfect. In other words, all numbers that fit that expression are perfect, while all even perfect numbers fit that form. This is due to the Euclid–Euler theorem, partially proved by Euclid and completed by Leonhard Euler: even numbers are perfect if and only if they can be expressed in the form 2 p − 1 × (2 p − 1), where 2 p − 1 is a Mersenne prime. There is a one-to-one correspondence between the Mersenne primes and the even perfect numbers. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. Meanwhile, perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. The numbers p corresponding to Mersenne primes must themselves be prime, although not all primes p lead to Mersenne primes-for example, 2 11 − 1 = 2047 = 23 × 89.

For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2 p − 1 for some positive integer p. Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Hence, there are 4 prime numbers between 20 and 40.Visualization of 6 as a perfect number Logarithmic graph of the number of digits of the largest known prime by year, nearly all of which have been Mersenne primes Prime numbers between 20 and 40 are 23, 29, 31, and 37.

Therefore, the average of prime numbers between 1 and 20 is 9.625.Įxample 3: Write the prime numbers between 20 and 40, and identify the count of these numbers? Therefore, the difference between the first and last 10 prime numbers is 603.Įxample 2: What is the average of the prime numbers between 1 and 20? Example 1: From 1 to 100, John counted the first 10 prime numbers and Max counted the last 10 prime numbers, what will be the difference between the sum of their count?įirst 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

List of prime numbers through 100