A prime number, or a prime, is a natural number which has exactly two distinct natural number divisors: 1 and itself. The number 1 is a special case which is considered neither prime nor composite. Although the number 1 used to be considered a prime it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own. With 1 excluded, the smallest prime is therefore 2. However, since 2 is the only even prime (which, ironically, in some sense makes it the “oddest” prime), it is also somewhat special, and the set of all primes excluding 2 is therefore called the “odd primes.” An infinite amount of prime numbers exists. The study of prime numbers is part of the number theory, the branch of mathematics, which encompasses the study of natural numbers. Prime numbers have been the subject of intense research, yet some fundamental questions have been unresolved for a long time. The problem of modeling the distribution of prime numbers is a popular subject of investigation for number theorists. When looking at individual numbers, the primes seem to be randomly distributed, but the global distribution of primes follows well-defined laws.
Properties of prime numbers:
- When written in base 10, all prime numbers except 2 and 5 end in 1, 3, 7 or 9. (Numbers ending in 0, 2, 4, 6 or 8 represent multiples of 2 and numbers ending in 0 or 5 represent multiples of 5.)
- If p is a prime number and p divides a product ab of integers, then p divides a or p divides b. This proposition was proved by Euclid and is known as Euclid’s lemma. It is used in some proofs of the uniqueness of prime factorizations.
- If p is prime and a is any integer, then ap − a is divisible by p
- An integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p (Wilson’s theorem). Conversely, an integer n > 4 is composite if and only if (n − 1)! is divisible by n.
- If n is a positive integer greater than 1, then there is always a prime number p with n < p < 2n
- Adding the reciprocals of all primes together results in a divergent infinite series (proof). More precisely, if S(x) denotes the sum of the reciprocals of all prime numbers p with p ≤ x, then S(x) = ln ln x + O(1) for x → ∞.
- All prime numbers above 3 are of the form 6n − 1 or 6n + 1, because all other numbers are divisible by 2 or 3. Generalizing this, all prime numbers above q are of form q#·n + m, where 0 < m < q, and m has no prime factor ≤ q.
“Prime Numbers.” Wikipedia. 15 Feb 2009.
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