17 Facts About Prime number

Prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.

The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen large number being prime is inversely proportional to its number of digits, that is, to its logarithm.

Yet another way to express the same thing is that a number is prime if it is greater than one and if none of the numbers divides evenly.

Therefore, every prime number other than 2 is an odd number, and is called an odd prime.

Prime number conjectured that all even perfect numbers come from Euclid's construction using Mersenne primes, but was unable to prove it.

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Christian Goldbach formulated Goldbach's conjecture, that every even number is the sum of two primes, in a 1742 letter to Euler.

Prime number introduced methods from mathematical analysis to this area in his proofs of the infinitude of the primes and the divergence of the sum of the reciprocals of the primes.

The fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with different numbers of copies of 1.

Conversely, if a number has the property that when it divides a product it always divides at least one factor of the product, then must be prime.

Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime .

The branch of Prime number theory studying such questions is called additive Prime number theory.

The original proof of the prime number theorem was based on a weak form of this hypothesis, that there are no zeros with real part equal to 1, although other more elementary proofs have been found.

Any other natural Prime number can be mapped into this system by replacing it by its remainder after division by.

Not every number that is prime among the integers remains prime in the Gaussian integers; for instance, the number 2 can be written as a product of the two Gaussian primes and.

Some fastest modern tests for whether an arbitrary given number is prime are probabilistic algorithms, meaning that they have a small random chance of producing an incorrect answer.

The Lucas–Lehmer primality test can determine whether a Mersenne number is prime, deterministically, in the same time as a single iteration of the Miller–Rabin test.

Concept of a prime number is so important that it has been generalized in different ways in various branches of mathematics.