PRIME NUMBERS

What are composite numbers?

A *composite number* is a positive integer that has *at least one positive divisor other than one or the number itself.*

What are prime numbers?

A *prime number* (or a prime) is a natural number greater than 1 that has *no positive divisors other than 1 and itself.*

What is a Mersenne prime?

A Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number that can be written in the form *M'n' = 2^n − 1* for some integer n.

What is a factorial prime?

A factorial prime is a prime number that is one less or one more than a factorial (all factorials > 1 are even).

What is a primality test?

A primality test is an algorithm for determining whether an input number is prime.

What is a prime gap?

A prime gap is the difference between two successive prime numbers.

What is a twin prime?

A twin prime is a prime number that has a prime gap of two.

When did Euclid discover the fact that there are infinitely many primes?

Around 300 B.C.

How many Mersenne primes are currently known?

As of September 2015, 48 Mersenne primes are known. The largest known prime number 2^57,885,161 − 1 is a Mersenne prime.

What is Euclid's proof of Euclid's theorem?

Consider any finite list of prime numbers p1, p2, ..., pn. Let P be the product of all the prime numbers in the list: P = p1p2...pn. Let q = P + 1. Then q is either prime or not: If q is prime, then there is at least one more prime than is in the list. If q is not prime, then some prime factor p divides q. If this factor p were on our list, then it would divide P (since P is the product of every number on the list); but p divides P + 1 = q. If p divides P and q, then p would have to divide the difference of the two numbers, which is (P + 1) − P or just 1. Since no prime number divides 1, this would be a contradiction and so p cannot be on the list. This means that at least one more prime number exists beyond those in the list.

What are cousin primes?

Cousin primes are prime numbers that differ by four.

What is the relationship between prime numbers and Euclid's lemma?

Euclid's lemma states that if a prime divides the product of two numbers, it must divide at least one of those numbers. *If p divides ab, then p divides either a or b.*

What is Euclid's theorem?

Euclid's theorem is a fundamental statement in number theory that asserts that *there are infinitely many prime numbers.*

Who demonstrated that there are infinitely many primes?

Euclid.

What is primality?

Primality is the property of being prime (or not).

What are sexy primes?

Sexy primes are prime numbers that differ from each other by six.

What is the prime number theorem?

The *prime number theorem* (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that *primes become less common as they become larger* by precisely quantifying the rate at which this occurs.

What is the relationship between prime numbers and the fundamental theorem of arithmetic?

The crucial importance of prime numbers to mathematics in general stems from the fundamental theorem of arithmetic, which states that every integer larger than 1 can be written as a product of one or more primes in a way that is unique except for the order of the prime factors.

What is the simplest primality test?

The simplest primality test is trial division: Given an input number n, check whether any prime integer m from 2 to √n evenly divides n (the division leaves no remainder). If n is divisible by any m then n is composite, otherwise it is prime.

What is the Great Internet Mersenne Prime Search (GIMPS)?

*GIMPS* is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers. The project was founded by *George Woltman*, who also wrote the software Prime95 and MPrime for the project.

What is PrimeGrid?

*PrimeGrid* is a distributed computing project for searching for prime numbers of world-record size. It makes use of the Berkeley Open Infrastructure for Network Computing (BOINC) platform. As of July 2014, there are over 14,400 active participants from 189 countries.