Prime number is a whole number greater than 1 that can be divided without a remainder only by itself and by 1. The sequence of prime numbers begins: 2, 3, 5, 7, 11, 13, 17, and so on. All whole numbers greater than 1 that are not prime numbers are called composite numbers.
Facts about primes.
The ancient Greek mathematician Euclid proved that every whole number greater than 1 can be written as the product of prime numbers in exactly one way. This fact is known as the fundamental theorem of arithmetic.
Euclid also showed that there must be infinitely many primes. If there were only a finite (limited) set of primes, a person could multiply all the primes together and add 1. This huge number would not be divisible by any prime in the set—including the prime that was supposedly the largest. It thus would either be divisible by an even larger prime, or it would be a larger prime itself.
Special primes.
Twin primes are prime numbers that differ by two. For example, 3 and 5 are twin primes, as are 41 and 43. A prime number, p, is said to be a Germain prime if 2p + 1 is also a prime number. Germain primes are named after the French mathematician Sophie Germain. Mersenne primes, named for the French mathematician Marin Mersenne, are of the form 2 p – 1, where the exponent p is also a prime number. Almost all of the largest known primes are Mersenne primes.
Extremely large primes are difficult to find, even with supercomputers. Mathematicians think there are infinitely many twin, Germain, and Mersenne primes. But they have not yet proven this idea.
See also Factor ; Number theory ; Sieve of Eratosthenes .
