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# Prime numbers

The prime numbers are the **"multiplicative atoms"** of the nonzero integers, whereas the composite numbers are the **"multiplicative molecules"** of the nonzero integers. The prime numbers are the integers which are divisible by one and only one nonunit (i.e. non-invertible integer) positive integer. The composite numbers are the integers which are divisible by more than one (but a finite number of) nonunit (i.e. non-invertible integer) positive integers.

Although 1 has been considered prime until the beginning of the 20^{th} century (former definition: no divisors apart from 1 and itself, itself not necessarily distinct from 1) the unit (i.e. multiplicatively invertible element) 1 is now widely known as the empty product (defined as the multiplicative identity, i.e. 1) of primes, where an integer is now considered prime iff it has exactly two divisors, a unit and a nonunit, where associates of (i.e. product of some unit with ) are not considered distinct divisors. This is the definition (resulting from the development of abstract algebra at the turn of the 20^{th} century) now accepted by most mathematicians.

## Contents

## Zero, units, primes and composites

Zero is divisible by all (infinite number of) nonzero integers (thus 0 is neither prime nor composite,) and it is also not the product of nonzero integers. Zero is also non-invertible (thus 0 is not a unit.)

A unit (i.e. invertible integer) is neither prime nor composite since it is divisible by no nonunit whatsoever, thus the units −1 and 1 of are neither prime nor composite.

The integers are either:

- Negative composite numbers: {−4, −6, −8, −9, −10, −12, −14, −15, −16, −18, −20, −21, −22, −24, −25, −26, −27, −28, ...} (Cf. −1 × A002808)
- Negative prime numbers: {−2, −3, −5, −7, −11, v13, −17, −19, −23, −29, −31, −37, −41, −43, −47, -53, -59, ...} (Cf. −1 × A000040)
- Negative unit: {−1}
- Zero: {0}
- Positive unit: {1}
- Positive primes numbers: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, ...} (Cf. A000040)
- Positive composite numbers: {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, ...} (Cf. A002808)

## Fundamental theorem of arithmetic

Fundamental theorem of arithmetic: Every integer can be written as a product of primes in an essentially unique way (up to units and ordering.)

## Infinitude of primes

In Book IX of the Elements, Euclid proved that there are infinitely many prime numbers (Cf. Euclid's proof that there are infinitely many primes.)

## Prime number theorem

## See also

- A008578 Prime numbers at the beginning of the 20
^{th}century (today 1 is no longer regarded as a prime, but as a unit). - A002808 The composite numbers: numbers of the form for and .

- Gaussian integers, Gaussian primes and Gaussian composites.
- Eisenstein integers, Eisenstein primes and Eisenstein composites.

## External links

- Eric W. Weisstein, Prime Number, from MathWorld — A Wolfram Web Resource..
- The Prime Pages (prime number research, records and resources)
- http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Prime_numbers.html
- Michael Coons, Yet another proof of the infinitude of primes, I.