prime factor

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 ${\displaystyle N}$ 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.

3. Used other than figuratively or idiomatically: see prime,‎ factor.
   • 1989 February 12, anonymous author, “Innocent Man In Prison Because He Is Gay”, in Gay Community News, volume 16, number 30, page 5:

     It became apparent to me that Mr. Genthner's open profession of being gay was the prime factor in his conviction and sentencing.

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