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There are known formulae to evaluate the prime-counting function (the number of primes smaller than a given value) faster than computing the primes. This has been used to compute that there are 1,925,320,391,606,803,968,923 primes (roughly 2 × 10 21 ) smaller than 10 23 .
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 ...
This means that, for n up to 2.5 × 10 10, if 2 n −1 (modulo n) equals 1, then n is prime, unless n is one of these 21853 pseudoprimes. Some composite numbers (Carmichael numbers) have the property that a n − 1 is 1 (modulo n) for every a that is coprime to n. The smallest example is n = 561 = 3·11·17, for which a 560 is 1 (modulo 561 ...
Inputs: n: a value to test for primality, n>3; k: a parameter that determines the number of times to test for primality Output: composite if n is composite, otherwise probably prime Repeat k times: Pick a randomly in the range [2, n − 2] If (), then return composite
1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310. We see that for composite n every term n# simply duplicates the preceding term (n − 1)#, as given in the definition. In the above example we have 12# = p 5 # = 11# since 12 is a composite number. Primorials are related to the first Chebyshev function, written ϑ(n) or θ(n) according to:
Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number. For example, 15 is a composite number because 15 = 3 · 5 , but 7 is a prime number because it cannot be decomposed in this way.
The first thousand values of φ(n).The points on the top line represent φ(p) when p is a prime number, which is p − 1. [1]In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n.
[1] [2] Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. [3] [4] E.g., the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7 but the integers 2 and 3 are not because each can only be divided by one and itself.