Integer factorization

Unsolved problem in computer science:

Can integer factorization be solved in polynomial time on a classical computer?

(more unsolved problems in computer science)

Innumber theory,integer factorizationis the decomposition of apositive integerinto aproductof integers. Every positive integer greater than 1 is either the product of two or more integerfactorsgreater than 1, in which case it is called acomposite number,or it is not, in which case it is called aprime number.For example, $15$ is a composite number because $15 = 3 \cdot 5$ ,but $7$ is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example $60 = 3 \cdot 20 = 3 \cdot (5 \cdot 4)$ .Continuing this process until every factor is prime is calledprime factorization;the result is always unique up to the order of the factors by theprime factorization theorem.

To factorize a small integer $n$ using mental or pen-and-paper arithmetic, the simplest method istrial division:checking if the number is divisible by prime numbers $2$ , $3$ , $5$ ,and so on, up to thesquare rootof $n$ .For larger numbers, especially when using a computer, various more sophisticated factorization algorithms are more efficient. A prime factorization algorithm typically involvestesting whether each factor is primeeach time a factor is found.

When the numbers are sufficiently large, no efficientnon-quantuminteger factorizationalgorithmis known. However, it has not been proven that such an algorithm does not exist. The presumeddifficultyof this problem is important for the algorithms used incryptographysuch asRSA public-key encryptionand theRSA digital signature.^[1]Many areas ofmathematicsandcomputer sciencehave been brought to bear on the problem, includingelliptic curves,algebraic number theory,andquantum computing.

Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) aresemiprimes,the product of two prime numbers. When they are both large, for instance more than two thousandbitslong, randomly chosen, and about the same size (but not too close, for example, to avoid efficient factorization byFermat's factorization method), even the fastest prime factorization algorithms on the fastest computers can take enough time to make the search impractical; that is, as the number of digits of the integer being factored increases, the number of operations required to perform the factorization on any computer increases drastically.

Many cryptographic protocols are based on the difficulty of factoring large composite integers or a related problem—for example, theRSA problem.An algorithm that efficiently factors an arbitrary integer would renderRSA-basedpublic-keycryptography insecure.

Prime decomposition

By thefundamental theorem of arithmetic,every positive integer has a uniqueprime factorization.(By convention, 1 is theempty product.)Testingwhether the integer is prime can be done inpolynomial time,for example, by theAKS primality test.If composite, however, the polynomial time tests give no insight into how to obtain the factors.

Given a general algorithm for integer factorization, any integer can be factored into its constituentprime factorsby repeated application of this algorithm. The situation is more complicated with special-purpose factorization algorithms, whose benefits may not be realized as well or even at all with the factors produced during decomposition. For example, if $n = 171 \times p \times q$ where $p < q$ are very large primes,trial divisionwill quickly produce the factors 3 and 19 but will take $p$ divisions to find the next factor. As a contrasting example, if $n$ is the product of the primes $13729$ , $1372933$ ,and $18848997161$ ,where $13729 \times 1372933 = 18848997157$ ,Fermat's factorization method will begin with $⌈ \sqrt n ⌉ = 18848997159$ which immediately yields $b = \sqrt a 2 - n = \sqrt 4 = 2$ and hence the factors $a - b = 18848997157$ and $a + b = 18848997161$ .While these are easily recognized as composite and prime respectively, Fermat's method will take much longer to factor the composite number because the starting value of $⌈ \sqrt 18848997157 ⌉ = 137292$ for $a$ is a factor of 10 from $1372933$ .

Current state of the art

Among the $b$ -bit numbers, the most difficult to factor in practice using existing algorithms are thosesemiprimeswhose factors are of similar size. For this reason, these are the integers used in cryptographic applications.

In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number (RSA-240) utilizing approximately 900 core-years of computing power.^[2]The researchers estimated that a 1024-bit RSA modulus would take about 500 times as long.^[3]

The largest such semiprime yet factored wasRSA-250,an 829-bit number with 250 decimal digits, in February 2020. The total computation time was roughly 2700 core-years of computing using IntelXeon Gold6130 at 2.1 GHz. Like all recent factorization records, this factorization was completed with a highly optimized implementation of thegeneral number field sieverun on hundreds of machines.

Time complexity

Noalgorithmhas been published that can factor all integers inpolynomial time,that is, that can factor a $b$ -bit number $n$ in time $O (b k)$ for some constant $k$ .Neither the existence nor non-existence of such algorithms has been proved, but it is generally suspected that they do not exist.^[4]^[5]

There are published algorithms that are faster than $O((1 + ε) b)$ for all positive $ε$ ,that is,sub-exponential.As of 2022^[update],the algorithm with best theoretical asymptotic running time is thegeneral number field sieve(GNFS), first published in 1993,^[6]running on a $b$ -bit number $n$ in time:

\exp \left(\left(\left({\tfrac {8}{3}}\right)^{\frac {2}{3}}+o(1)\right)\left(\log n\right)^{\frac {1}{3}}\left(\log \log n\right)^{\frac {2}{3}}\right).

For current computers, GNFS is the best published algorithm for large $n$ (more than about 400 bits). For aquantum computer,however,Peter Shordiscovered an algorithm in 1994 that solves it in polynomial time.Shor's algorithmtakes only $O(b 3)$ time and $O(b)$ space on $b$ -bit number inputs. In 2001, Shor's algorithm was implemented for the first time, by usingNMRtechniques on molecules that provide seven qubits.^[7]

In order to talk aboutcomplexity classessuch as P, NP, and co-NP, the problem has to be stated as adecision problem.

Decision problem(Integer factorization)—For every natural numbers $n$ and $k$ ,does $n$ have a factor smaller than $k$ besides 1?

It is known to be in bothNPandco-NP,meaning that both "yes" and "no" answers can be verified in polynomial time. An answer of "yes" can be certified by exhibiting a factorization $n=d(.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠n/d⁠)$ with $d \leq k$ .An answer of "no" can be certified by exhibiting the factorization of $n$ into distinct primes, all larger than $k$ ;one can verify their primality using theAKS primality test,and then multiply them to obtain $n$ .Thefundamental theorem of arithmeticguarantees that there is only one possible string of increasing primes that will be accepted, which shows that the problem is in bothUPand co-UP.^[8]It is known to be inBQPbecause of Shor's algorithm.

The problem is suspected to be outside all three of the complexity classes P, NP-complete,^[9]andco-NP-complete. It is therefore a candidate for theNP-intermediatecomplexity class.

In contrast, the decision problem "Is $n$ a composite number? "(or equivalently:" Is $n$ a prime number? ") appears to be much easier than the problem of specifying factors of $n$ .The composite/prime problem can be solved in polynomial time (in the number $b$ of digits of $n$ ) with theAKS primality test.In addition, there are severalprobabilistic algorithmsthat can test primality very quickly in practice if one is willing to accept a vanishingly small possibility of error. The ease ofprimality testingis a crucial part of theRSAalgorithm, as it is necessary to find large prime numbers to start with.

Factoring algorithms

Special-purpose

A special-purpose factoring algorithm's running time depends on the properties of the number to be factored or on one of its unknown factors: size, special form, etc. The parameters which determine the running time vary among algorithms.

An important subclass of special-purpose factoring algorithms is theCategory 1orFirst Categoryalgorithms, whose running time depends on the size of smallest prime factor. Given an integer of unknown form, these methods are usually applied before general-purpose methods to remove small factors.^[10]For example, naivetrial divisionis a Category 1 algorithm.

Trial division
Wheel factorization
Pollard's rho algorithm,which has two common flavors toidentify group cycles:one by Floyd and one by Brent.
Algebraic-group factorization algorithms,among which arePollard's $p - 1$ algorithm,Williams' $p + 1$ algorithm,andLenstra elliptic curve factorization
Fermat's factorization method
Euler's factorization method
Special number field sieve
Difference of two squares

General-purpose

A general-purpose factoring algorithm, also known as aCategory 2,Second Category,orKraitchikfamilyalgorithm,^[10]has a running time which depends solely on the size of the integer to be factored. This is the type of algorithm used to factorRSA numbers.Most general-purpose factoring algorithms are based on thecongruence of squaresmethod.

Other notable algorithms

Shor's algorithm,for quantum computers

Heuristic running time

In number theory, there are many integer factoring algorithms that heuristically have expectedrunning time

L_{n}\left[{\tfrac {1}{2}},1+o(1)\right]=e^{(1+o(1)){\sqrt {(\log n)(\log \log n)}}}

inlittle-oandL-notation. Some examples of those algorithms are theelliptic curve methodand thequadratic sieve. Another such algorithm is theclass group relations methodproposed by Schnorr,^[11]Seysen,^[12]and Lenstra,^[13]which they proved only assuming the unprovedgeneralized Riemann hypothesis.

Rigorous running time

The Schnorr–Seysen–Lenstra probabilistic algorithm has been rigorously proven by Lenstra and Pomerance^[14]to have expected running time $L n [⁠ 1 / 2 ⁠,1+ o (1)]$ by replacing the GRH assumption with the use of multipliers. The algorithm uses theclass groupof positive binaryquadratic formsofdiscriminant $Δ$ denoted by $G Δ$ . $G Δ$ is the set of triples of integers $(a, b, c)$ in which those integers are relative prime.

Schnorr–Seysen–Lenstra algorithm

Given an integer $n$ that will be factored, where $n$ is an odd positive integer greater than a certain constant. In this factoring algorithm the discriminant $Δ$ is chosen as a multiple of $n$ , $Δ = - dn$ ,where $d$ is some positive multiplier. The algorithm expects that for one $d$ there exist enoughsmoothforms in $G Δ$ .Lenstra and Pomerance show that the choice of $d$ can be restricted to a small set to guarantee the smoothness result.

Denote by $P Δ$ the set of all primes $q$ withKronecker symbol $(⁠ Δ / q ⁠) = 1$ .By constructing a set ofgeneratorsof $G Δ$ and prime forms $f q$ of $G Δ$ with $q$ in $P Δ$ a sequence of relations between the set of generators and $f q$ are produced. The size of $q$ can be bounded by $c 0 (log| Δ |) 2$ for some constant $c 0$ .

The relation that will be used is a relation between the product of powers that is equal to theneutral elementof $G Δ$ .These relations will be used to construct a so-called ambiguous form of $G Δ$ ,which is an element of $G Δ$ of order dividing 2. By calculating the corresponding factorization of $Δ$ and by taking agcd,this ambiguous form provides the complete prime factorization of $n$ .This algorithm has these main steps:

Let $n$ be the number to be factored.

Let $Δ$ be a negative integer with $Δ = - dn$ ,where $d$ is a multiplier and $Δ$ is the negative discriminant of some quadratic form.
Take the $t$ first primes $p 1 = 2, p 2 = 3, p 3 = 5,..., p t$ ,for some $t \in N$ .
Let $f q$ be a random prime form of $G Δ$ with $(⁠ Δ / q ⁠) = 1$ .
Find a generating set $X$ of $G Δ$ .
Collect a sequence of relations between set $X$ and ${f q : q \in P Δ}$ satisfying:
$\left(\prod _{x\in X_{}}x^{r(x)}\right).\left(\prod _{q\in P_{\Delta }}f_{q}^{t(q)}\right)=1.$
Construct an ambiguous form $(a, b, c)$ that is an element $f \in G Δ$ of order dividing 2 to obtain a coprime factorization of the largest odd divisor of $Δ$ in which $Δ = -4 ac$ or $Δ = a (a - 4 c)$ or $Δ = (b - 2 a)(b + 2 a)$ .
If the ambiguous form provides a factorization of $n$ then stop, otherwise find another ambiguous form until the factorization of $n$ is found. In order to prevent useless ambiguous forms from generating, build up the2-Sylowgroup $Sll 2 (Δ)$ of $G (Δ)$ .

To obtain an algorithm for factoring any positive integer, it is necessary to add a few steps to this algorithm such as trial division, and theJacobi sum test.

Expected running time

The algorithm as stated is aprobabilistic algorithmas it makes random choices. Its expected running time is at most $L n [⁠ 1 / 2 ⁠,1+ o (1)]$ .^[14]

Notes

^Lenstra, Arjen K. (2011),"Integer Factoring",in van Tilborg, Henk C. A.; Jajodia, Sushil (eds.),Encyclopedia of Cryptography and Security,Boston, MA: Springer US, pp. 611–618,doi:10.1007/978-1-4419-5906-5_455,ISBN 978-1-4419-5905-8,retrieved2022-06-22
^"[Cado-nfs-discuss] 795-bit factoring and discrete logarithms".Archived fromthe originalon 2019-12-02.
^Kleinjung, Thorsten; Aoki, Kazumaro; Franke, Jens; Lenstra, Arjen K.; Thomé, Emmanuel; Bos, Joppe W.; Gaudry, Pierrick; Kruppa, Alexander; Montgomery, Peter L.; Osvik, Dag Arne; te Riele, Herman J. J.; Timofeev, Andrey; Zimmermann, Paul (2010)."Factorization of a 768-Bit RSA Modulus"(PDF).In Rabin, Tal (ed.).Advances in Cryptology - CRYPTO 2010, 30th Annual Cryptology Conference, Santa Barbara, CA, USA, August 15-19, 2010. Proceedings.Lecture Notes in Computer Science. Vol. 6223. Springer. pp. 333–350.doi:10.1007/978-3-642-14623-7_18.
^Krantz, Steven G.(2011),The Proof is in the Pudding: The Changing Nature of Mathematical Proof,New York: Springer, p. 203,doi:10.1007/978-0-387-48744-1,ISBN 978-0-387-48908-7,MR 2789493
^Arora, Sanjeev;Barak, Boaz (2009),Computational complexity,Cambridge: Cambridge University Press, p. 230,doi:10.1017/CBO9780511804090,ISBN 978-0-521-42426-4,MR 2500087,S2CID 215746906
^Buhler, J. P.; Lenstra, H. W. Jr.; Pomerance, Carl (1993). "Factoring integers with the number field sieve".The development of the number field sieve.Lecture Notes in Mathematics. Vol. 1554. Springer. pp. 50–94.doi:10.1007/BFb0091539.hdl:1887/2149.ISBN 978-3-540-57013-4.Retrieved12 March2021.
^ Vandersypen, Lieven M. K.; et al. (2001). "Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance".Nature.414(6866): 883–887.arXiv:quant-ph/0112176.Bibcode:2001Natur.414..883V.doi:10.1038/414883a.PMID 11780055.S2CID 4400832.
^ Lance Fortnow (2002-09-13)."Computational Complexity Blog: Complexity Class of the Week: Factoring".
^Goldreich, Oded;Wigderson, Avi(2008), "IV.20 Computational Complexity", inGowers, Timothy;Barrow-Green, June;Leader, Imre(eds.),The Princeton Companion to Mathematics,Princeton, New Jersey: Princeton University Press, pp. 575–604,ISBN 978-0-691-11880-2,MR 2467561.See in particularp. 583.
^^a ^b David BressoudandStan Wagon(2000).A Course in Computational Number Theory.Key College Publishing/Springer. pp.168–69.ISBN 978-1-930190-10-8.
^Schnorr, Claus P. (1982)."Refined analysis and improvements on some factoring algorithms".Journal of Algorithms.3(2): 101–127.doi:10.1016/0196-6774(82)90012-8.MR 0657269.Archived fromthe originalon September 24, 2017.
^Seysen, Martin (1987)."A probabilistic factorization algorithm with quadratic forms of negative discriminant".Mathematics of Computation.48(178): 757–780.doi:10.1090/S0025-5718-1987-0878705-X.MR 0878705.
^Lenstra, Arjen K (1988)."Fast and rigorous factorization under the generalized Riemann hypothesis"(PDF).Indagationes Mathematicae.50(4): 443–454.doi:10.1016/S1385-7258(88)80022-2.
^^a ^bLenstra, H. W.; Pomerance, Carl (July 1992)."A Rigorous Time Bound for Factoring Integers"(PDF).Journal of the American Mathematical Society.5(3): 483–516.doi:10.1090/S0894-0347-1992-1137100-0.MR 1137100.

References

Richard CrandallandCarl Pomerance(2001).Prime Numbers: A Computational Perspective.Springer.ISBN 0-387-94777-9.Chapter 5: Exponential Factoring Algorithms, pp. 191–226. Chapter 6: Subexponential Factoring Algorithms, pp. 227–284. Section 7.4: Elliptic curve method, pp. 301–313.
Donald Knuth.The Art of Computer Programming,Volume 2:Seminumerical Algorithms,Third Edition. Addison-Wesley, 1997.ISBN 0-201-89684-2.Section 4.5.4: Factoring into Primes, pp. 379–417.
Samuel S. Wagstaff Jr.(2013).The Joy of Factoring.Providence, RI: American Mathematical Society.ISBN 978-1-4704-1048-3..
Warren, Henry S. Jr. (2013).Hacker's Delight(2 ed.).Addison Wesley-Pearson Education, Inc.ISBN 978-0-321-84268-8.

External links

msieve– SIQS and NFS – has helped complete some of the largest public factorizations known
Richard P. Brent, "Recent Progress and Prospects for Integer Factorisation Algorithms",Computing and Combinatorics ",2000, pp. 3–22.download
Manindra Agrawal,Neeraj Kayal, Nitin Saxena, "PRIMES is in P." Annals of Mathematics 160(2): 781–793 (2004).August 2005 version PDF
Eric W. Weisstein,“RSA-640 Factored”MathWorld Headline News,November 8, 2005
Dario Alpern's Integer factorization calculator– A web app for factoring large integers

[1] Lenstra, Arjen K. (2011),"Integer Factoring",in van Tilborg, Henk C. A.; Jajodia, Sushil (eds.),Encyclopedia of Cryptography and Security,Boston, MA: Springer US, pp. 611–618,doi:10.1007/978-1-4419-5906-5_455,ISBN 978-1-4419-5905-8,retrieved2022-06-22

[2] "[Cado-nfs-discuss] 795-bit factoring and discrete logarithms".Archived fromthe originalon 2019-12-02.

[rsa768-3] Kleinjung, Thorsten; Aoki, Kazumaro; Franke, Jens; Lenstra, Arjen K.; Thomé, Emmanuel; Bos, Joppe W.; Gaudry, Pierrick; Kruppa, Alexander; Montgomery, Peter L.; Osvik, Dag Arne; te Riele, Herman J. J.; Timofeev, Andrey; Zimmermann, Paul (2010)."Factorization of a 768-Bit RSA Modulus"(PDF).In Rabin, Tal (ed.).Advances in Cryptology - CRYPTO 2010, 30th Annual Cryptology Conference, Santa Barbara, CA, USA, August 15-19, 2010. Proceedings.Lecture Notes in Computer Science. Vol. 6223. Springer. pp. 333–350.doi:10.1007/978-3-642-14623-7_18.

[4] Krantz, Steven G.(2011),The Proof is in the Pudding: The Changing Nature of Mathematical Proof,New York: Springer, p. 203,doi:10.1007/978-0-387-48744-1,ISBN 978-0-387-48908-7,MR 2789493

[5] Arora, Sanjeev;Barak, Boaz (2009),Computational complexity,Cambridge: Cambridge University Press, p. 230,doi:10.1017/CBO9780511804090,ISBN 978-0-521-42426-4,MR 2500087,S2CID 215746906

[6] Buhler, J. P.; Lenstra, H. W. Jr.; Pomerance, Carl (1993). "Factoring integers with the number field sieve".The development of the number field sieve.Lecture Notes in Mathematics. Vol. 1554. Springer. pp. 50–94.doi:10.1007/BFb0091539.hdl:1887/2149.ISBN 978-3-540-57013-4.Retrieved12 March2021.

[7] Vandersypen, Lieven M. K.; et al. (2001). "Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance".Nature.414(6866): 883–887.arXiv:quant-ph/0112176.Bibcode:2001Natur.414..883V.doi:10.1038/414883a.PMID 11780055.S2CID 4400832.

[8] Lance Fortnow (2002-09-13)."Computational Complexity Blog: Complexity Class of the Week: Factoring".

[9] Goldreich, Oded;Wigderson, Avi(2008), "IV.20 Computational Complexity", inGowers, Timothy;Barrow-Green, June;Leader, Imre(eds.),The Princeton Companion to Mathematics,Princeton, New Jersey: Princeton University Press, pp. 575–604,ISBN 978-0-691-11880-2,MR 2467561.See in particularp. 583.

[Bressoud_and_Wagon-10] David BressoudandStan Wagon(2000).A Course in Computational Number Theory.Key College Publishing/Springer. pp.168–69.ISBN 978-1-930190-10-8.

[1982-schnorr-11] Schnorr, Claus P. (1982)."Refined analysis and improvements on some factoring algorithms".Journal of Algorithms.3(2): 101–127.doi:10.1016/0196-6774(82)90012-8.MR 0657269.Archived fromthe originalon September 24, 2017.

[1987-seysen-12] Seysen, Martin (1987)."A probabilistic factorization algorithm with quadratic forms of negative discriminant".Mathematics of Computation.48(178): 757–780.doi:10.1090/S0025-5718-1987-0878705-X.MR 0878705.

[1988-lenstra-13] Lenstra, Arjen K (1988)."Fast and rigorous factorization under the generalized Riemann hypothesis"(PDF).Indagationes Mathematicae.50(4): 443–454.doi:10.1016/S1385-7258(88)80022-2.

[lenstra-pomerance-14] Lenstra, H. W.; Pomerance, Carl (July 1992)."A Rigorous Time Bound for Factoring Integers"(PDF).Journal of the American Mathematical Society.5(3): 483–516.doi:10.1090/S0894-0347-1992-1137100-0.MR 1137100.

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v t e Computational hardness assumptions
Number theoretic	Integer factorization Phi-hiding RSA problem Strong RSA Quadratic residuosity Decisional composite residuosity Higher residuosity
Group theoretic	Discrete logarithm Diffie-Hellman Decisional Diffie–Hellman Computational Diffie–Hellman
Pairings	External Diffie–Hellman Sub-group hiding Decision linear
Lattices	Shortest vector problem(gap) Closest vector problem(gap) Learning with errors Ring learning with errors Short integer solution
Non-cryptographic	Exponential time hypothesis Unique games conjecture Planted clique conjecture

v t e Number-theoretic algorithms
Primality tests	AKS APR Baillie–PSW Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius Solovay–Strassen Miller–Rabin
Prime-generating	Sieve of Atkin Sieve of Eratosthenes Sieve of Pritchard Sieve of Sundaram Wheel factorization
Integer factorization	Continued fraction (CFRAC) Dixon's Lenstra elliptic curve (ECM) Euler's Pollard's rho p− 1 p+ 1 Quadratic sieve (QS) General number field sieve (GNFS) Special number field sieve (SNFS) Rational sieve Fermat's Shanks's square forms Trial division Shor's
Multiplication	Ancient Egyptian Long Karatsuba Toom–Cook Schönhage–Strassen Fürer's
Euclidean division	Binary Chunking Fourier Goldschmidt Newton-Raphson Long Short SRT
Discrete logarithm	Baby-step giant-step Pollard rho Pollard kangaroo Pohlig–Hellman Index calculus Function field sieve
Greatest common divisor	Binary Euclidean Extended Euclidean Lehmer's
Modular square root	Cipolla Pocklington's Tonelli–Shanks Berlekamp Kunerth
Other algorithms	Chakravala Cornacchia Exponentiation by squaring Integer square root Integer relation(LLL;KZ) Modular exponentiation Montgomery reduction Schoof Trachtenberg system
Italicsindicate that algorithm is for numbers of special forms

v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
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(Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Descartes Refactorable Superperfect