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The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range. Interface to a list of the first 98 million primes (primes less than 2,000,000,000) Weisstein, Eric W. "Prime Number Sequences". MathWorld. Selected prime related sequences in OEIS.
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 ...
The first such distribution found is π(N) ~ N / log(N) , where π(N) is the prime-counting function (the number of primes less than or equal to N) and log(N) is the natural logarithm of N. This means that for large enough N , the probability that a random integer not greater than N is prime is very close to 1 / log( N ) .
However, it does not contain all the prime numbers, since the terms gcd(n + 1, a n) are always odd and so never equal to 2. 587 is the smallest prime (other than 2) not appearing in the first 10,000 outcomes that are different from 1. Nevertheless, in the same paper it was conjectured to contain all odd primes, even though it is rather inefficient.
Any prime number is prime to any number it does not measure. [note 6] Proposition 30 If two numbers, by multiplying one another, make the same number, and any prime number measures the product, it also measures one of the original numbers. [note 7] Proof of 30 If c, a prime number, measure ab, c measures either a or b. Suppose c does not measure a.
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. [1] [2] It is denoted by π(x) (unrelated to the number π). A symmetric variant seen sometimes is π 0 (x), which is equal to π(x) − 1 ⁄ 2 if x is exactly a prime number, and equal to π(x) otherwise.
If either p n # + 1 or p n # − 1 is a primorial prime, it means that there are larger primes than the nth prime (if neither is a prime, that also proves the infinitude of primes, but less directly; each of these two numbers has a remainder of either p − 1 or 1 when divided by any of the first n primes, and hence all its prime factors are ...
Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2 p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1.