Prime numbers

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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 $\scriptstyle n \,$ is now considered prime iff it has exactly two divisors, a unit and a nonunit, where associates of $\scriptstyle k \,$ (i.e. product of some unit $\scriptstyle u \,$ with $\scriptstyle k \,$ ) 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.

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 $\scriptstyle \mathbb{Z}\,$ 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)