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Accomplishments in factoring large integers
Integer factorization is the process of determining which prime numbers divide a given positive integer. Doing this quickly has applications in cryptography
Integer_factorization_records
Decomposition of a number into a product
prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer n using
Integer_factorization
Set of large semiprimes
decimal digits (330 bits). Its factorization was announced on April 1, 1991, by Arjen K. Lenstra. Reportedly, the factorization took a few days using the multiple-polynomial
RSA_numbers
Prime number of the form 2^n – 1
special number field sieve that allows the factorization of several numbers at once. See Integer factorization records for links to more information. The special
Mersenne_prime
Complex number whose real and imaginary parts are both integers
unique factorization and many related properties. However, Gaussian integers do not have a total order that respects arithmetic. Gaussian integers are algebraic
Gaussian_integer
Challenge for factoring large semiprimes
another RSA challenge posed in 1977 RSA Secret-Key Challenge Integer factorization records Jevon's number Kaliski, Burt (18 Mar 1991). "Announcement of
RSA_Factoring_Challenge
Natural number
This factorization also makes it a highly powerful number, a number with a record-setting value of the product of the exponents in its factorization. Among
288_(number)
American mathematician (1947–2020)
field sieve methods of factorization; he has been involved in the computations which set a number of integer factorization records. He was a Putnam Fellow
Peter Montgomery (mathematician)
Peter_Montgomery_(mathematician)
listed below. List of largest known primes and probable primes Integer factorization records "GIMPS Project Discovers Largest Known Prime Number: 2136,279
Largest_known_prime_number
Approach to public-key cryptography
used in several integer factorization algorithms that have applications in cryptography, such as Lenstra elliptic-curve factorization. The use of elliptic
Elliptic-curve_cryptography
Special-purpose integer factorization algorithm
integer factorization algorithm. The general number field sieve (GNFS) was derived from it. The special number field sieve is efficient for integers of
Special_number_field_sieve
Integer factorization algorithm
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second-fastest method known (after the general number field
Quadratic_sieve
A prime p divides a^p–a for any integer a
same difficulty as the factorization of n, since φ(n) = (p − 1)(q − 1), and conversely, the factors p and q are the (integer) solutions of the equation
Fermat's_little_theorem
Natural number
digits. "Facts about the integer". mathworld.wolfram.com. Sloane, N. J. A. (ed.). "Sequence A100827 (Highly cototient numbers: records for a(n) in A063741)"
299_(number)
Natural number
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A100827 (Highly cototient numbers: records for a(n) in A063741
6000_(number)
Natural number
record value of the product of the exponents in prime factorization)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-13
1728_(number)
Mathematical polynomial formula
in elementary algebra. Binomial numbers generalize this factorization to higher odd powers. Starting with the expression, a 2 − a b + b 2
Sum_of_two_cubes
Natural number
parity. Thus reduction modulo 2 records the parity of an integer: even integers are congruent to 0 modulo 2, and odd integers are congruent to 1 modulo 2
2
Algorithm for generating numbers coprime with first few primes
thus be used for an improvement of the trial division method for integer factorization, as none of the generated numbers need be tested in trial divisions
Wheel_factorization
Type of mathematical expression
form, called factorization is, in general, too difficult to be done by hand-written computation. However, efficient polynomial factorization algorithms
Polynomial
Natural number
fourth safe prime. 23 is the first prime p for which unique factorization of cyclotomic integers based on the pth root of unity breaks down. 23 is the smallest
23_(number)
Number divisible only by 1 and itself
Prime factors calculator can factorize any positive integer up to 20 digits. Fast Online primality test with factorization makes use of the Elliptic Curve
Prime_number
Natural number
1867. In computing, this number is the largest value that a signed 32-bit integer field can hold. At the time of its discovery, 2,147,483,647 was the largest
2,147,483,647
Natural number
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A100827 (Highly cototient numbers: records for a(n) in A063741)"
1000_(number)
Topics referred to by the same term
involved in a multiplication Divisor, an integer which evenly divides a number without leaving a remainder Factorization, the decomposition of an object into
Factor
Natural number
a number, numeral, and grapheme. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property
1
Natural number
only prime triangular number, and Carl Friedrich Gauss proved that every integer is the sum of at most three triangular numbers. Three is the only prime
3
Integer side lengths of a right triangle
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), a well-known
Pythagorean_triple
Numbers that contain only the digit 1
Another special situation is b = −4k4, with k positive integer, which has the aurifeuillean factorization, for example, b = −4 (with k = 1, then R2 and R3 are
Repunit
Branch of pure mathematics
composite numbers. Factorization is a method of expressing a number as a product. Specifically in number theory, integer factorization is the decomposition
Number_theory
Mechanism for authenticating cryptographic keys
PGP/GPG Keys and fingerprints can also be added into a server's DNSSEC DNS records. So any users who want to communicate securely (or any software users)
Web_of_trust
Number of integers coprime to and less than n
greatest common divisor of n and every positive integer less than n, which suffices to provide the factorization anyway. The property established by Gauss,
Euler's_totient_function
American mathematician, cryptologist and computer scientist (born 1971)
integer factorization: a proposal". cr.yp.to. Arjen K. Lenstra; Adi Shamir; Jim Tomlinson; Eran Tromer (2002). "Analysis of Bernstein's Factorization
Daniel_J._Bernstein
Non-federated cryptographic protocol
The first version of the protocol, TextSecure v1, was based on Off-the-record messaging (OTR). On 24 February 2014, Open Whisper Systems introduced TextSecure
Signal_Protocol
Key agreement protocol
consisting of a private key d {\displaystyle d} (a randomly selected integer in the interval [ 1 , n − 1 ] {\displaystyle [1,n-1]} ) and a public key
Elliptic-curve_Diffie–Hellman
Natural number
(151 + 157). It is the totient sum of the first 41 integers. 309 = 3 × 103. It is a Blum integer and a centered icosahedral number. 310 = 2 × 5 × 31
300_(number)
write the prime factorization of n in base 10 and concatenate the factors; iterate until a prime is reached. For a non-negative integer, its home prime
List_of_prime_numbers
Natural number
points on a 6*6*6 cube is 152. PRP records Sloane, N. J. A. (ed.). "Sequence A005897". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Wikimedia
152_(number)
Positive integer of the form (2^(2^n))+1
them, is a positive integer of the form: F n = 2 2 n + 1 , {\displaystyle F_{n}=2^{2^{n}}+1,} where n is a non-negative integer. The first few Fermat
Fermat_number
Natural number between 89 and 91
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-11. Sloane, N. J. A. (ed.). "Sequence A200503 (Record (maximal) gaps between
90_(number)
1-factorable. The perfect 1-factorization conjecture that every complete graph on an even number of vertices admits a perfect 1-factorization. Cereceda's conjecture
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Unique positive real number which when multiplied by itself gives 2
reasonably small denominator. Sequence A002193 in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square
Square_root_of_2
Group obtained by aggregating similar elements of a larger group
the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of n {\displaystyle
Quotient_group
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences
45_(number)
eπ163=262537412640768743.99999999999925…, is an almost integer, differing from the nearest integer by approximately 7.5×10−13. (0.000000000001; 1000−4;
Orders_of_magnitude_(numbers)
Analytic function in mathematics
positive integers is zero. The Euler product formula can be used to calculate the asymptotic probability that s randomly selected integers within a bound
Riemann_zeta_function
Cryptographic key management algorithm
Algorithm's design is based on the DH ratchet that was introduced by Off-the-Record Messaging (OTR) and combines it with a symmetric-key ratchet modeled after
Double_Ratchet_Algorithm
Natural number
The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Kourbatov, Alexei (16 September 2013). "Tables of record gaps between prime constellations"
100,000
Natural number
earliest phonograph records turns 78 revolutions per minute. "Sloane's A007304 : Sphenic numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
78_(number)
Topics referred to by the same term
interaction in natural and synthetic populations Quadratic sieve, an integer factorization algorithm Quality Score, a variable used by search engines to set
QS
Composite number in number theory
relation: b n ≡ b ( mod n ) {\displaystyle b^{n}\equiv b{\pmod {n}}} for all integers b {\displaystyle b} . The relation may also be expressed in the form:
Carmichael_number
Natural number
a record that stands until 197. 114 is the smallest positive integer which has yet to be represented as a3 + b3 + c3, where a, b, and c are integers. It
114_(number)
Mathematical scheme for verifying the authenticity of digital documents
Digitalized Signatures and Public Key Functions as Intractable as Factorization (PDF) (Technical report). Cambridge, MA, United States: MIT Laboratory
Digital_signature
Natural number
has a reduced totient of 500, and a totient of 4,000, with a total of 16 integers having a totient value of 10,000. There are a total of 1,229 prime numbers
10,000
Natural number
definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers via the 15 and 290 theorems
15_(number)
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A259180 (Amicable pairs.)". The On-Line Encyclopedia of Integer Sequences
500_(number)
Swedish mathematician (1929–2014)
April 2025. Caldwell, Chris K. (2013). "The Top Twenty: Record Prime Progressions". Journal of Integer Sequences. 16 (1). Article 13.1.6. Section 4 gives a
Hans_Riesel
Natural number
than 101th, is the correct form. 101 is a prime number and the smallest integer above 100. It is also a palindromic number, and hence, a palindromic prime
101_(number)
Best results achieved to date
current record, published in 2022, is over G F ( 2111023 50 ) {\displaystyle \mathrm {GF} (2111023^{50})} .[citation needed] Previous records for integers modulo
Discrete_logarithm_records
Natural number
trees on 6 labeled nodes 4356 = 662, sum of the cubes of the first eleven integers 4359 – perfect totient number 4369 – seventh super-Poulet number 4371 –
4000_(number)
Branch of elementary mathematics
restricted to the study of integers and focuses on their properties and relationships such as divisibility, factorization, and primality. Traditionally
Arithmetic
Two consecutive integers for which the sums of the prime factors of each are equal
a factorization. The name was given by Carl Pomerance for Babe Ruth and Hank Aaron, as Ruth's career regular-season home run total was 714, a record which
Ruth–Aaron_pair
Natural number
the number of divisors of n (A000005), increases to a record.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-19. "A002201
12_(number)
Natural number
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. "Sloane's A005277 : Nontotients". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
318_(number)
Texas Instruments' response to a project to factorize cryptographic keys
te Riele (1999-08-26), New factorization record Archived 2021-06-24 at the Wayback Machine (announcement of factorization of RSA-155). Retrieved on 2008-03-10
Texas Instruments signing key controversy
Texas_Instruments_signing_key_controversy
Count of the possible partitions of a set
of Integer Sequences. 11 (2): Article 08.2.5, 3. Bibcode:2008JIntS..11...25S. MR 2420912. Wagstaff, Samuel S. (1996). "Aurifeuillian factorizations and
Bell_number
Natural number
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. "Sloane's A051634 : Strong primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
127_(number)
Positive real number which when multiplied by itself gives 5
shown to be Euclidean, and hence a unique factorization domain, by Richard Dedekind. This is the ring of integers in the golden field Q ( 5 ) {\displaystyle
Square_root_of_5
Infinite sum that is considered independently from any notion of convergence
{\displaystyle R[x],} in the same way as the p-adic integers are the p-adic completion of the ring of the integers.) Formal powers series in several indeterminates
Formal_power_series
Natural number
the number of divisors of n (A000005), increases to a record.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31. "A002201
360_(number)
Natural number
+ 2 are 3 consecutive Harshad numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. "Official Count to 1 Trillion Thread". v t
511_(number)
Probabilistic algorithm for computing discrete logarithms
empty_list for k = 1 , 2 , … {\displaystyle k=1,2,\ldots } Using an integer factorization algorithm optimized for smooth numbers, try to factor g k mod q
Index_calculus_algorithm
Study of analyzing information systems in order to discover their hidden aspects
constructed problems in pure mathematics, the best-known being integer factorization. In encryption, confidential information (called the "plaintext")
Cryptanalysis
Natural number
2021-10-06. "Sloane's A076980 : Leyland numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-02. "A061209 - OEIS". oeis
512_(number)
Natural number
16-bit integer on a computer. 32768 = 215 = 85 = 323, maximum absolute value of a negative value for a signed (two's complement) 16-bit integer on a computer
30,000
Natural number
(ed.). "Sequence A007770 (Happy numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A051400 (Smallest
219_(number)
Composite number that passes Fermat's probable primality test
divisible by p {\displaystyle p} . For a positive integer a {\displaystyle a} , if a composite integer x {\displaystyle x} divides a x − 1 − 1 {\displaystyle
Fermat_pseudoprime
Function in number theory
without performing factorization along the way. A further extension is the Kronecker symbol, in which the bottom argument may be any integer. The power residue
Legendre_symbol
Numbers obtained by adding the two previous ones
Fibonacci numbers Fn are: The Fibonacci sequence can be extended to negative integer indices by following the same recurrence relation in the negative direction
Fibonacci_sequence
Numeral system
chosen due to its prime factorization: 2×2×3×5, which makes it divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Integers and fractions were represented
Babylonian_cuneiform_numerals
Natural number
hundred [and] sixty or twelve hundred [and] sixty) is the natural number and integer following 1259 and preceding 1261. 1260 is a highly composite number, a
1260_(number)
Number of the form x^y + y^x
number of the form x y + y x {\displaystyle x^{y}+y^{x}} where x and y are integers greater than 1. They are named after the mathematician Paul Leyland. The
Leyland_number
Natural number
145–159. "Sloane's A007770 : Happy numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved June 1, 2016. "A000959 Lucky numbers
13_(number)
Natural number
Adria is a large main belt asteroid 143 Records label of producer David Foster, a sub-label of Atlantic Records Psalm 143 Sonnet 143 by William Shakespeare
143_(number)
Algebraic curve in mathematics
also find applications in elliptic curve cryptography (ECC) and integer factorization. An elliptic curve is not an ellipse in the sense of a projective
Elliptic_curve
field sieve Lenstra elliptic curve factorization Pollard's p − 1 algorithm Pollard's rho algorithm prime factorization algorithm Quadratic sieve Shor's
List_of_algorithms
German mental calculist and mathematician
also calculated a 7-digit logarithm table and extended a table of integer factorizations from 6,000,000 to 9,000,000. Dase had very little knowledge of mathematical
Zacharias_Dase
Conjecture on zeros of the zeta function
that the Riemann zeta function has its zeros only at the negative even integers and at complex numbers with real part 12. Many consider it to be the most
Riemann_hypothesis
Composite number which passes Miller–Rabin primality test
Fermat pseudoprime but not a strong pseudoprime, this even gives us a factorization: 31697 = gcd(28419+1, 31697) × gcd(28419−1, 31697) = 29 × 1093. For
Strong_pseudoprime
Short sequence of bytes used to authenticate or look up a longer public key
fingerprint (or the key it refers to) will be stored locally along with a record of the other user's name or address, so that future communications with
Public_key_fingerprint
Natural number
function)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A058763 (Integers which are neither totient nor cototient)"
218_(number)
Equation for radii of tangent circles
0\leq 2m\leq d\leq e} . Factorizations of sums of two squares can be obtained using the sum of two squares theorem. Any other integer Apollonian gasket can
Descartes's_theorem
Certain constant-recursive integer sequences
n ( P , Q ) {\displaystyle V_{n}(P,Q)} are certain constant-recursive integer sequences that satisfy the recurrence relation x n = P ⋅ x n − 1 − Q ⋅
Lucas_sequence
State University, Fullerton Integer factorization Performs parts of the number field sieve in the factorization of large integers Yes 4,096 (Mar 2023) 50
List of volunteer computing projects
List_of_volunteer_computing_projects
Standardized tests
to select multiple correct responses. Arithmetic: Integer, Divisibility rule, Integer factorization, Prime number, Remainder, Parity Exponentiation, nth
Graduate_Record_Examinations
Cryptography secured against quantum computers
rely on the difficulty of one of three mathematical problems: the integer factorization problem, the discrete logarithm problem, or the elliptic-curve discrete
Post-quantum_cryptography
Natural number
(A000005), increases to a record)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. "The Positive Integer 840". www.positiveintegers
840_(number)
Prime pair of the form (p, 2p+1)
system being broken by some factorization algorithms such as Pollard's p − 1 algorithm. However, with the current factorization technology, the advantage
Safe and Sophie Germain primes
Safe_and_Sophie_Germain_primes
Quality of zero being an even number
parity—the quality of an integer being even or odd—is even. This can be easily verified based on the definition of "even": zero is an integer multiple of 2, specifically
Parity_of_zero
Method of data analysis
components transformation can also be associated with another matrix factorization, the singular value decomposition (SVD) of X, 𝐗=𝐔𝜮𝐖T Here Σ is an
Principal_component_analysis
travel, tourism, insurance
INTEGER FACTORIZATION-RECORDS
INTEGER FACTORIZATION-RECORDS
INTEGER FACTORIZATION-RECORDS
INTEGER FACTORIZATION-RECORDS
INTEGER FACTORIZATION-RECORDS
INTEGER FACTORIZATION-RECORDS
INTEGER FACTORIZATION-RECORDS
INTEGER FACTORIZATION-RECORDS
INTEGER FACTORIZATION-RECORDS
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