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A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a ...
Except for 2 and 5, all prime numbers end in 1, 3, 7, or 9. Dirichlet's theorem states that asymptotically, 25% of all primes end in each of these four digits. However, empirical evidence shows that the number of primes that end in 3 or 7 less than n tends to be slightly bigger than the number of primes that end in 1 or 9 less than n (a ...
Since each natural number greater than 1 has at least one prime factor, and two successive numbers n and (n + 1) have no factor in common, the product n(n + 1) has more different prime factors than the number n itself. So the chain of pronic numbers:
A simple formula is. for positive integer , where is the floor function, which rounds down to the nearest integer. By Wilson's theorem, is prime if and only if . Thus, when is prime, the first factor in the product becomes one, and the formula produces the prime number . But when is not prime, the first factor becomes zero and the formula ...
An odd prime number p is defined to be regular if it does not divide the class number of the pth cyclotomic field Q(ζ p), where ζ p is a primitive pth root of unity. The prime number 2 is often considered regular as well. The class number of the cyclotomic field is the number of ideals of the ring of integers Z(ζ p) up to equivalence.
This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes.
The prime number theorem asserts that an integer m selected at random has roughly a 1 / ln m chance of being prime. Thus if n is a large even integer and m is a number between 3 and n / 2 , then one might expect the probability of m and n − m simultaneously being prime to be 1 / ln m ln(n − m) .
John Selfridge has conjectured that if p is an odd number, and p ≡ ±2 (mod 5), then p will be prime if both of the following hold: 2 p−1 ≡ 1 (mod p), f p+1 ≡ 0 (mod p), where f k is the k-th Fibonacci number. The first condition is the Fermat primality test using base 2.