prime factor

prime factor (plural prime factors)

1. (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 ${\displaystyle N}$ chosen at random having a prime factor between ${\displaystyle N^{\alpha }}$ and ${\displaystyle N^{\alpha (1+\delta )}}$ is approximately ${\displaystyle =\delta }$, independent of the magnitude of ${\displaystyle \alpha }$, if ${\displaystyle \delta }$ 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.

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.

• prime factorization

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

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