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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.
If a number m can be expressed as a string of prime length to some base, such a number may or may not be prime, but commonly is not; for example, to base 10, there are only three such numbers of length less than 100 (1 is by definition, not prime). The three are: 11 (length 2), 1111111111111111111 (length 19), and 11111111111111111111111 ...
Therefore, every prime number other than 2 is an odd number, and is called an odd prime. [10] Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in ...
For example, π(10) = 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that x / log x is a good approximation to π(x) (where log here means the natural logarithm), in the sense that the limit of the quotient of the two functions π(x) and x / log x as x increases without ...
The number x = 2 is most often used in this basic primality check, and n = 341 = 11 × 31 is notable since , and n = 341 is the smallest composite number for which x = 2 is a false witness to primality.
In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1. [2] [3] The integers k of this form are sometimes referred to as totatives of n. For example, the totatives of n = 9 are the six numbers 1, 2, 4, 5, 7 and 8.
When k is equal to n, the value cannot be prime since n 2 − n + n = n 2 is divisible by n. Since the polynomial can be written as k(k−1) + n, using the integers k with −(n−1) < k ≤ 0 produces the same set of numbers as 1 ≤ k < n. These polynomials are all members of the larger set of prime generating polynomials.
Only repunits (in any base) having a prime number of digits can be prime. This is a necessary but not sufficient condition. For example, R 11 (2) = 2 11 − 1 = 2047 = 23 × 89. If p is an odd prime, then every prime q that divides R p (b) must be either 1 plus a multiple of 2p, or a factor of b − 1. For example, a prime factor of R 29 is ...