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...or of two integers involves [[Unique factorization|factoring]] both into [[prime number]]s: Observe that the prime number 2 occurs twice in the factorization of 60, and three times in the factoriza

...theorem the "prime number theorem" and a different prime number theorem a "Prime Number Theorem" (due to the latter being mentioned in a title). In the case of "fu ...per noun because we say '''the''' Prime Number Theorem rather than '''a''' prime number theorem.

{{r|Prime Number Theorem}} {{r|Prime number}}

...meter and number theorist [[Euclid]] of [[Alexandria]], states that if a [[prime number]] ''p'' is a [[divisor]] of the [[multiplication|product]] of two [[integer

...rime''' is a composite number that has certain properties in common with [[prime number]]s. ...can test it for properties that all prime numbers share. One property of a prime number is that it is only divisible by one and itself. This is a defining property

* If <math> p^n </math> is a power of a prime number, then

...'except for 2 itself''. Then multiples of 3, 5, 7, 11, and so on for every prime number less than or equal to the ''square root'' of ''N'', are crossed out. In thi

{{r|Prime number}}

A prime number that divides a product of two integers must divide one of the two integers.

* [[Paulo Ribenboim]]: The New Book of Prime Number Records. Springer-Verlag, 1996, ISBN 0-387-94457-5

...ed that if ''n'' is a whole number for which <math> 2^{n} - 1 </math> is [[prime number|prime]], then the number

...nly 23% of the two-digit numbers and 16% of the three-digit numbers. The [[Prime number#Locating primes|trial division method]] provides an intuitive explanation. ...the [[natural logarithm]] of <math>n</math>). The formal statement of the Prime Number Theorem is

A natural number (except 1) which has no proper divisor is called ''[[prime number|prime]]'', * If a prime number divides a product ''ab'', and it does not divide ''a'', then it divides ''b

* [[Paulo Ribenboim]]. The New Book of Prime Number Records. Springer-Verlag, 1996, ISBN 0-387-94457-5

* [[Paulo Ribenboim]]. The New Book of Prime Number Records. Springer-Verlag, 1996, ISBN 0-387-94457-5

* [[Paulo Ribenboim]], ''The New Book of Prime Number Records'' (3 ed.), Springer, 1996, ISBN 0-387-94457-5

* Similarly, a [[finite field]] of [[prime number|prime]] order has only the identity automorphism. ...> of prime power order ''q'', where <math>q = p^f</math> is a power of the prime number ''p'', has the [[Frobenius automorphism]], <math>\Phi: x \mapsto x^p</math>

...s of the module can be constructed. This is analogous to viewing as the [[prime number]]s as building blocks from which all [[whole number]]s can be assembled. L

* <math>p^n</math> and <math>2p^n</math> where ''p'' is an odd [[prime number|prime]].

*If <math>p\ </math> is a prime number, than <math>p\ </math> divides <math>L_p - 1\ </math>. The converse is fals