prime factor

English

Noun

prime factor (plural prime factors)

2. (number theory) A factor of a given integer which is also a prime number.
   • 1847, James Robinson, The American Arithmetic, John P. Jewett & Co., page 91:

     Hence, the prime factors of 100 are 1, 2, 2, 5, 5, and 1 × 2 × 2 × 5 × 5 = 100. From the above illustration, we derive the following rule for finding all the prime factors of any composite number.

   • 1994, Hans Riesel, Prime Numbers and Computer Methods for Factorization, Springer, page 161:

     Theorem 5.5 Dickman's Theorem. The probability of a number $N$ chosen at random having a prime factor between $N^{\alpha }$ and $N^{\alpha (1+\delta )}$ is approximately $=\delta$, independent of the magnitude of $\alpha$, if $\delta$ is small.

   • 2003, Gary R. Jensen, Arithmetic for Teachers: With Applications and Topics from Geometry, American Mathematical Society, page 180:

     If any subset of the set of prime factors of a be taken, then the product of the elements of this subset is a factor of a.
     For example, 12 is a factor of 60 and the set of prime factors of 12 is {2, 2, 3}, which is a subset of {2, 2, 3, 5}, which is the set of prime factors of 60.

Translations

factor

• Chinese:

  Mandarin: 質因數／质因数 (zh) (zhìyīnshù)

• Esperanto: prima faktoro, prima faktorigo
• Finnish: alkutekijä (fi)
• French: facteur premier (fr) m
• German: Primfaktor (de) m
• Indonesian: faktor prima
• Irish: fachtóir príomha m
• Italian: fattore primo (it) m
• Japanese: 素因数 (ja) (soinsū)
• Malay: faktor perdana
• Manx: ard-faghteyr m
• Maori: tauwehe toitū
• Norwegian: primfaktor m
• Polish: czynnik pierwszy m
• Romanian: factor prim (ro) m
• Russian: просто́й дели́тель (prostój delítelʹ)

Verb

prime factor (third-person singular simple present prime factors, present participle prime factoring, simple past and past participle prime factored)

1. (transitive) To reduce an integer to its set of prime factors.
   • 2004, Fred N. Grayson, CliffsTestPrep Military Flight Aptitude Tests, Wiley, page 39:

     Any composite number can be prime factored; that is, it can be written as a product of prime numbers (excluding 1) in one and only one way. […] When prime factoring numbers, it is standard to rearrange the factors so that the numbers are in increasing order.

   • 2008, Phil Pine, Peterson's Master the SAT 2009, page 322:

     For example, the number 30, which is not prime, can be prime factored as 2 x 3 x 5.

   • 2009, Jerome E. Kaufmann, Karen L. Schwitters, Elementary Algebra, 9th edition, Cengage Learning, page 386:

     Another variation of the technique for changing radicals to simplest form is to prime factor the radicand and then to look for perfect squares in exponential form.

Related terms

• prime factorization

Translations

Translations

• Finnish: jakaa alkutekijöihin
• French: décomposer en facteurs premiers
• Polish: rozkładać na czynniki pierwsze impf, rozłożyć na czynniki pierwsze pf

See also

• factorization
• prime decomposition
• prime element
• prime number
• unique factorization ring

Further reading

• Prime number on Wikipedia.Wikipedia
• Fundamental theorem of arithmetic on Wikipedia.Wikipedia
• Irreducible polynomial on Wikipedia.Wikipedia
• Factorization of polynomials on Wikipedia.Wikipedia