Mersenne primes M p are also noteworthy due to their connection with perfect numbers . In the 4th century BC, Euclid proved that if 2 p − 1 is prime , then 2 p − 1 ...

A boy has four red marbles and eight blue marbles. He arranges his twelve marbles randomly, in a ring. What is the probability that no two red marbles are adjacent?

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Prime and Composite : Any integer which is divisible by 1 and itself only is called a prime number.

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The third Venn diagram below is a little different. This time you need to decide what sets the two groups represent. Choose the labels from the list on the right.

Each of Webster's definitions may be applied to this page, but
the most operative is 2a: * An integer greater than one
is prime if its only positive divisors are itself and one*
(otherwise it is composite ). For example: 15 is composite
because it has the two prime divisors 3 and 5.

If you just need a way to generate very big prime numbers and don't care to generate all prime numbers < an integer n, you can use Lucas-Lehmer test to verify Mersenne prime numbers. A Mersenne prime number is in the form of 2^p -1. I think that Lucas-Lehmer test is the fastest algorithm discovered for Mersenne prime numbers.

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