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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 20th 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 20th 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:

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

Prime number theorem

See also

  • A008578 Prime numbers at the beginning of the 20th century (today 1 is no longer regarded as a prime, but as a unit).
  • A002808 The composite numbers: numbers \scriptstyle n\, of the form \scriptstyle xy\, for \scriptstyle x \,>\, 1\, and \scriptstyle y \,>\, 1\,.

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