Prime factor

This image demonstrates how to find the prime factorization of 864. A shorthand way of writing the resulting prime factors is 2⁵ × 3³

In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly.^[1] The prime factorization of a positive integer is a list of the integer's prime factors, together with their multiplicities; the process of determining these factors is called integer factorization. The fundamental theorem of arithmetic says that every positive integer has a single unique prime factorization.^[2]

To shorten prime factorizations, factors are often expressed in powers (multiplicities). For example,

360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 = 2^3 \times 3^2 \times 5,

in which the factors 2, 3 and 5 have multiplicities of 3, 2 and 1, respectively.

For a prime factor p of n, the multiplicity of p is the largest exponent a for which p^a divides n exactly.

For a positive integer n, the number of prime factors of n and the sum of the prime factors of n (not counting multiplicity) are examples of arithmetic functions of n that are additive but not completely additive.^[3]

Perfect squares

Perfect square numbers can be recognized by the fact that all of their prime factors have even multiplicities. For example, the number 144 (the square of 12) has the prime factors

144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^4 \times 3^2.

These can be rearranged to make the pattern more visible:

144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = (2 \times 2 \times 3) \times (2 \times 2 \times 3) = (2 \times 2 \times 3)^2 = (12)^2.

Because every prime factor appears an even number of times, the original number can be expressed as the square of some smaller number. In the same way, perfect cube numbers will have prime factors whose multiplicities are multiples of three, and so on.

Coprime integers

Positive integers with no prime factors in common are said to be coprime. Two integers a and b can also be defined as coprime if their greatest common divisor gcd(a, b) = 1. Euclid's algorithm can be used to determine whether two integers are coprime without knowing their prime factors; the algorithm runs in a time that is polynomial in the number of digits involved.

The integer 1 is coprime to every positive integer, including itself. This is because it has no prime factors; it is the empty product. This implies that gcd(1, b) = 1 for any b ≥ 1.

Cryptographic applications

Determining the prime factors of a number is an example of a problem frequently used to ensure cryptographic security in encryption systems;^[4] this problem is believed to require super-polynomial time in the number of digits — it is relatively easy to construct a problem that would take longer than the known age of the universe to solve on current computers using current algorithms.

Omega functions

The function, $ω (n)$ (omega), represents the number of distinct prime factors of n, while the function, $Ω(n)$ (big omega), represents the number of prime factors of $n$ counted by multiplicities.^[2] If

n=\prod _{i=1}^{\omega (n)}p_{i}^{\alpha _{i}},

then

\Omega (n)=\sum _{i=1}^{\omega (n)}\alpha _{i}.

For example, $24 = 2 3 \times 3 1$ , so $ω (24) = 2$ and $Ω(24) = 3 + 1 = 4$ .

$ω (n)$ for $n$ = 1, 2, 3, … is respectively 0, 1, 1, 1, 1, 2, 1, 1, 1, … (sequence A001221 in the OEIS).
$Ω(n)$ for $n$ = 1, 2, 3, … is respectively 0, 1, 1, 2, 1, 2, 1, 3, 2, … (sequence A001222 in the OEIS).

References

↑ Jensen, Gary R. (2004). Arithmetic for Teachers: With Applications and Topics from Geometry. American Mathematical Society.
1 2 Riesel, Hans (1994), Prime numbers and computer methods for factorization, Basel, Switzerland: Birkhäuser, ISBN 978-0-8176-3743-9
↑ Melvyn B. Nathanson (1996). Additive Number Theory: the Classical Bases. Graduate Texts in Mathematics. 234. Springer-Verlag. ISBN 0-387-94656-X.
↑ Menezes, Alfred; van Oorschot, Paul C.; Vanstone, Scott A. (October 1996). Handbook of Applied Cryptography. CRC Press. ISBN 0-8493-8523-7.

Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable Sociable Betrothed
Other sets	Deficient Friendly Solitary Sublime Harmonic divisor Frugal Equidigital Extravagant

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