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Algorithm for checking if a number is prime
In computational number theory, the Lucas test is a primality test for a natural number n; it requires that the prime factors of n − 1 be already known
Lucas_primality_test
Test if a Mersenne number is prime
mathematics, the Lucas–Lehmer test (LLT) is a primality test for Mersenne numbers. The test was originally developed by Édouard Lucas in 1878 and subsequently
Lucas–Lehmer_primality_test
Algorithm for determining whether a number is prime
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike
Primality_test
Probabilistic primality test
The Miller–Rabin primality test or Rabin–Miller primality test is a probabilistic primality test: an algorithm which determines whether a given number
Miller–Rabin_primality_test
Topics referred to by the same term
Lucas test may refer to Lucas primality test for primality of general numbers Lucas–Lehmer primality test for Mersenne primes Lucas' reagent, used to
Lucas_test
Probabilistic primality testing algorithm
primality test? More unsolved problems in mathematics The Baillie–PSW primality test is a probabilistic or possibly deterministic primality testing algorithm
Baillie–PSW_primality_test
Number-theoretic algorithm
{\displaystyle N} is prime. It produces a primality certificate to be found with less effort than the Lucas primality test, which requires the full factorization
Pocklington_primality_test
French mathematician (1842–1891)
Lehmer refined Lucas's primality tests and obtained the Lucas–Lehmer primality test. He worked on the development of the umbral calculus. Lucas is credited
Édouard_Lucas
Probabilistic primality test
is the number of times we test a random a, and n is the value we want to test for primality; see Miller–Rabin primality test for details. There are infinitely
Fermat_primality_test
Proof that a number is prime
science, a primality certificate or primality proof is a succinct, formal proof that a number is prime. Primality certificates allow the primality of a number
Primality_certificate
Primality test for certain numbers
In mathematics, the Lucas–Lehmer–Riesel test is a primality test for numbers of the form N = k · 2n − 1 with odd k < 2n. The test was developed by Hans
Lucas–Lehmer–Riesel_test
Number divisible only by 1 and itself
called primality. A simple but slow method of checking the primality of a given number n {\displaystyle n} , called trial division, tests whether
Prime_number
Algorithm checking for prime numbers
The AKS primality test (also known as the Agrawal–Kayal–Saxena primality test and the cyclotomic AKS test) is a deterministic primality-proving algorithm
AKS_primality_test
Methods to test or prove primality
curve primality testing techniques, or elliptic curve primality proving (ECPP), are among the quickest and most widely used methods in primality proving
Elliptic_curve_primality
A prime p divides a^p–a for any integer a
This theorem forms the basis for the Lucas primality test, an important primality test, and Pratt's primality certificate. If a and p are coprime numbers
Fermat's_little_theorem
Probabilistic test for the primality of an integer
more stringent primality test than equation (1). There are infinitely many strong Lucas pseudoprimes, and therefore, infinitely many Lucas pseudoprimes
Lucas_pseudoprime
Freeware application to search for primes
prime"). For much of its history, it used the Lucas–Lehmer primality test, but the availability of Lucas–Lehmer assignments was deprecated in April 2021
Prime95
Topics referred to by the same term
urinary stones LLT GM High Feature engine, a type of engine Lucas–Lehmer primality test for Mersenne numbers Cholesky decomposition, an algorithm to
LLT
Integers that satisfy a specific condition
Provable prime Baillie–PSW primality test Euler–Jacobi pseudoprime Lucas pseudoprime Miller–Rabin primality test Perrin primality test Carmichael number The
Probable_prime
Primality tests: determining whether a given number is prime AKS primality test Baillie–PSW primality test Fermat primality test Lucas primality test
List_of_algorithms
Baillie–PSW primality test Miller–Rabin primality test Lucas–Lehmer primality test Lucas–Lehmer test for Mersenne numbers AKS primality test Pollard's p − 1
List_of_number_theory_topics
Mathematical conjectures about Mersenne primes
distribution of numbers of prime factors of Mersenne numbers Lucas–Lehmer primality test Lucas primality test Catalan's Mersenne conjecture Mersenne's laws Bateman
Mersenne_conjectures
Prime number of the form 2^n – 1
test to determine whether a given Mersenne number is prime: the Lucas–Lehmer primality test (LLT), which makes it much easier to test the primality of
Mersenne_prime
American mathematician (born 1945)
Samuel S. Wagstaff, Jr. (July 2021). "Strengthening the Baillie-PSW Primality Test" (PDF). Mathematics of Computation. 90 (330): 1931–1955. arXiv:2006
Samuel_S._Wagstaff_Jr.
Certain constant-recursive integer sequences
Baillie–PSW primality test. Lucas sequences are used in some primality proof methods, including the Lucas–Lehmer and Lucas–Lehmer–Riesel tests and the hybrid
Lucas_sequence
Number sequence 3,0,2,3,2,5,5,7,10,...
In contrast, the Lucas pseudoprimes are anti-correlated. Presumably, combining the Perrin and Lucas tests should make a primality test as strong as the
Perrin_number
Type of pseudoprime
the Miller–Rabin primality test), 1.5 times that of a Lucas pseudoprimality test, and slightly more than a Baillie–PSW primality test. Note that the quadratic
Frobenius_pseudoprime
Positive integer of the form (2^(2^n))+1
on Fermat primes. Double exponential function Lucas' theorem Mersenne prime Pierpont prime Primality test Proth's theorem Pseudoprime Sierpiński number
Fermat_number
Probable prime that is composite
instead of primes. On the other hand, deterministic primality tests, such as the AKS primality test, do not give false positives; because of this, there
Pseudoprime
Decomposition of a number into a product
digits of n) with the AKS primality test. In addition, there are several probabilistic algorithms that can test primality very quickly in practice if
Integer_factorization
Type of odd composite number
Jacobi symbol. Unlike the standard Lucas pseudoprimes, there is no known efficient primality test using the Lucas d-pseudoprimes. Hence they are not generally
Somer–Lucas_pseudoprime
Generalization of the Legendre symbol in number theory
theory, but its main use is in computational number theory, especially primality testing and integer factorization; these in turn are important in cryptography
Jacobi_symbol
Surname list
Lucas–Lehmer primality test, in mathematics, after Derrick Henry Lehmer Lehmer random number generator, named after D. H. Lehmer Lehmer sieve Lucas–Lehmer
Lehmer
test (QFT) is a probabilistic primality test to determine whether a number is a probable prime. It is named after Ferdinand Georg Frobenius. The test
Quadratic_Frobenius_test
Composite number which passes Miller–Rabin primality test
is a composite number that passes the Miller–Rabin primality test. All prime numbers pass this test, but a small fraction of composites also pass, making
Strong_pseudoprime
Topics referred to by the same term
Research Lloyd's Law Reports Log-likelihood ratio Lucas–Lehmer–Riesel test, an algorithm to find the primality of a number of the form k*2n-1 Lunar laser ranging
LLR
Volunteer project using software to search for Mersenne prime numbers
project relied primarily on the Lucas–Lehmer primality test (LL), an algorithm that is both specialized for testing Mersenne primes and particularly
Great Internet Mersenne Prime Search
Great_Internet_Mersenne_Prime_Search
Ancestor of all current life on Earth
generally thought to share common ancestry. On the basis of a formal statistical test, this theory of a universal common ancestry (UCA) is supported in preference
Last universal common ancestor
Last_universal_common_ancestor
Prime number of the form k*(2^n)+1
093322456 for the reciprocal sum of Proth numbers. The primality of Proth numbers can be tested more easily than many other numbers of similar magnitude
Proth_prime
the discovery. New Mersenne primes are found using the Lucas–Lehmer test (LLT), a primality test for Mersenne primes that is efficient for binary computers
List of Mersenne primes and perfect numbers
List_of_Mersenne_primes_and_perfect_numbers
Wiki-based programming chrestomathy
Look-and-say sequence Lucas numbers Lucas–Lehmer primality test Mandelbrot set (draw) Mersenne primes Miller–Rabin primality test Morse code Numerical
Rosetta_Code
Composite number that passes Fermat's probable primality test
numbers is to generate random odd numbers and test them for primality. However, deterministic primality tests are slow. If the user is willing to tolerate
Fermat_pseudoprime
Number of form 2^(2^p-1)-1 with prime exponent
discovered this sequence after the discovery of the primality of M 127 = c 4 {\displaystyle M_{127}=c_{4}} by Lucas in 1876.p. 22 Catalan conjectured that they
Double_Mersenne_number
Algorithmic runtime requirements for common math procedures
hdl:21.11116/0000-0005-717D-0. Tao, Terence (2010). "1.11 The AKS primality test". An epsilon of room, II: Pages from year three of a mathematical blog
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
been proved prime by computer with a primality test for their form, for example the Lucas–Lehmer primality test for Mersenne numbers. “!” is the factorial
List of largest known primes and probable primes
List_of_largest_known_primes_and_probable_primes
1911 silent film by D. W. Griffith
The Primal Call is a 1911 American short silent romance film directed by D. W. Griffith, starring Wilfred Lucas and featuring Blanche Sweet. Wilfred Lucas
The_Primal_Call
Canadian mathematician (born 1943)
the history of primality tests. In it, he showed among other things that Édouard Lucas worked shortly before his early death on a test similar to today's
Hugh_C._Williams
American mathematician (1905–1991)
computational number theory. Lehmer refined Édouard Lucas' work in the 1930s and devised the Lucas–Lehmer test for Mersenne primes. His peripatetic career as
D._H._Lehmer
Composite number in number theory
hold. This fact precludes the use of that theorem as an absolute test of primality. The Carmichael numbers form the subset K1 of the Knödel numbers.
Carmichael_number
Number of the form x^y + y^x
largest prime whose primality was proved by elliptic curve primality proving. In December 2012, this was improved by proving the primality of the two numbers
Leyland_number
Prime number of the form 2^u × 3^v + 1
thus its primality can be tested by Proth's theorem. On the other hand, when 2 u < 3 v {\displaystyle 2^{u}<3^{v}} alternative primality tests for M =
Pierpont_prime
Odd composite number which passes the given congruence
equation can be tested rather quickly, which can be used for probabilistic primality testing. These tests are twice as strong as tests based on Fermat's
Euler_pseudoprime
American mathematician (1875–1952)
Obituary by D. H. Lehmer Hugh C. Williams (1998). Édouard Lucas and Primality Testing. Wiley. ISBN 978-0-471-14852-4. The Prime Pages website Mersenne
Ralph_Ernest_Powers
Odd composite number which passes the given congruence
equation can be tested rather quickly, which can be used for probabilistic primality testing. These tests are over twice as strong as tests based on Fermat's
Euler–Jacobi_pseudoprime
American mathematician (1911–1995)
early computers to obtain results. For example, he coded the Lucas–Lehmer primality test to determine whether 2n − 1 was prime for all prime n < 2304
Raphael_M._Robinson
Numbers obtained by adding the two previous ones
-~\!\left({\frac {5}{p}}\right)}.} The above formula can be used as a primality test in the sense that if n ∣ F n − ( 5 n ) , {\displaystyle n\mid F_{n\
Fibonacci_sequence
Russian-born American animator (born 1970)
(For Programming Less Than One Hour)" in 2004. Star Wars creator George Lucas hired Tartakovsky to direct Star Wars: Clone Wars (2003–2005), an animated
Genndy_Tartakovsky
Polyhedral number representing a tetrahedron
Fermat pseudoprime Frobenius pseudoprime Lucas pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic
Tetrahedral_number
Numbers that contain only the digit 1
though Édouard Lucas showed no prime below three million had period nineteen, there was no attempt to test any repunit for primality until early in the
Repunit
Prime number that is product of first n primes ± 1
pn# is the primorial of pn (i.e. the product of the first n primes). Primality tests show that: pn# − 1 is prime for n = 2, 3, 5, 6, 13, 24, 66, 68, 167
Primorial_prime
Type of computer benchmarking tool
SPECfp is a computer benchmark designed to test the floating-point performance of a computer. It is managed by the Standard Performance Evaluation Corporation
SPECfp
Integer having a non-trivial divisor
called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, which do
Composite_number
than a power of two, because they can be verified by a specialized primality test that is faster than the general one. As of October 2024[update], the
Largest_known_prime_number
Season of television series
and she enlists Hopper's help in uncovering the cause. While Mike and Lucas Sinclair attempt to repair the former's relationship with Eleven, Will Byers
Stranger_Things_season_3
Mathematics conference
proposals. The Édouard Lucas Memorial Lecturer, Carl Pomerance, spoke on "Primality testing - variations on a theme of Lucas." The 12th ICFNTA took place
International Conference on Fibonacci Numbers and their Applications
International_Conference_on_Fibonacci_Numbers_and_their_Applications
2016 video game
PC Preview". IGN. Retrieved 2023-04-07. Thomas, Lucas M. (2016-02-22). "Out This Week: Far Cry Primal, Superhot, Pokémon Red/Blue/Yellow". IGN. Retrieved
We_Are_the_Dwarves
American mass media conglomerate (2025–present)
from the original on April 7, 2025. Retrieved April 8, 2025. Manfredi, Lucas (June 5, 2025). "Former FCC Commissioner Warns Paramount Will Be 'Melting
Paramount_Skydance
Comic book series by DC Comics
Snyder and Dragotta Talk "Absolute Batman"". DC. Retrieved March 31, 2026. Lucas, Bryant (October 11, 2024). "Spotlight Review: Absolute Batman #1". Dark
Absolute_Batman
2002 film by Guillermo del Toro
gifted new technician. A pandemic is turning vampires into "Reapers", primal, mutant creatures with a ravenous thirst for blood and a highly infectious
Blade_II
Pseudoscientific analysis of handwriting
of 4.0 or higher: Luscher Color Test, Szondi Test, handwriting analysis (graphology), Bender Visual Motor Gestalt Test (for assessment of neuropsychological
Graphology
Mathematical sequences
the Lucas sequence is the second row. See also Fibonacci integer sequences modulo n. A different generalization of the Fibonacci sequence is the Lucas sequences
Generalizations of Fibonacci numbers
Generalizations_of_Fibonacci_numbers
Fortnite, Towelie was also added as a sidekick. January 11: Season 3 of Primal premiered on Adult Swim. January 15: A third Phineas and Ferb film is announced
2026_in_animation
Oriolo Film Studios, Inc. Japanese co-production —N/a Traditional Johnny Test 6 117 2005–2014 The WB (Kids' WB) (2005–2008) Cartoon Network (2009–2014)
List of American animated television series
List_of_American_animated_television_series
2023 film by Steven Caple Jr.
interview with Noah due to having talked with Noah's former commanding officer. Lucas Huarancca as Amaru, the descendant of a tribe that helped the Maximals for
Transformers: Rise of the Beasts
Transformers:_Rise_of_the_Beasts
Prime number of the form (2ᵖ+1)/3
"An efficient probable prime test for numbers of the form (2p + 1)/3". Tony Reix, "Three conjectures about primality testing for Mersenne, Wagstaff and
Wagstaff_prime
Karen; Antonoglou, Ioannis; Huang, Aja; Guez, Arthur; Hubert, Thomas; Baker, Lucas; Lai, Matthew; Bolton, Adrian; Chen, Yutian; Lillicrap, Timothy; Fan, Hui;
Timeline of artificial intelligence
Timeline_of_artificial_intelligence
is programmed to destroy all of Project X's test subjects, starting with Wolverine. Another Project X test subject, Silver Fox, steps forward to interrogate
List of Marvel Comics characters: P
List_of_Marvel_Comics_characters:_P
On TV, Files For Chapter 11 Bankruptcy". Deadline Hollywood. Manfredi, Lucas; Ray, Alyssa (April 17, 2026). "Nexstar-Tegna Merger Put on Ice by Federal
2026_in_American_television
Netflix media franchise
Things: Lucas on the Line by Suyi Davies Okungbowa – Published on July 26, 2022, this novel is set after the third season and told from Lucas Sinclair's
Stranger_Things_(franchise)
Share New Album Chapter 1". Stereogum. Retrieved January 9, 2026. Burrows, Luca (January 15, 2026). "Zu – Ferrum Sidereum". The Quietus. Retrieved January
List_of_2026_albums
11th 0 3 Army Girl 1938 11th 0 3 Carefree 1938 11th 0 3 Suez 1938 11th 0 3 Test Pilot 1938 11th 0 3 Blockade 1938 11th 0 2 Sweethearts 1938 11th 0 (1) 2
List of Academy Award–nominated films
List_of_Academy_Award–nominated_films
Leonard Adleman – RSA, DNA computing Manindra Agrawal – polynomial-time primality testing Luis von Ahn – human-based computation Alfred Aho – compilers book
List_of_computer_scientists
Number, approximately 2.20557
pseudoprime' as defined in: Adams, William; Shanks, Daniel (1982). "Strong primality tests that are not sufficient". Mathematics of Computation. 39 (159). American
Supersilver_ratio
2018 video game
be used to halt the Cleansing. When Lara saves a boy named Etzli (Kamran Lucas), she and Jonah are brought into Paititi by his mother Unuratu (Patricia
Shadow_of_the_Tomb_Raider
English artist (born 1963)
basement and studio space where Emin and Lucas worked. Emin and Lucas were able to fund The Shop with money Lucas had from Charles Saatchi. The Shop was
Tracey_Emin
Japanese actor (1920–1997)
(1967). According to his daughter, he also turned down an offer from George Lucas to play either Darth Vader or Obi-Wan Kenobi in Star Wars (1977). Mifune
Toshiro_Mifune
24–28. Tasuku Hatanaka Voice actor. Won the title in a side-step stamina test. Pre-recorded between February 4–14; aired between February 24–28. Tetsu
2025 in professional wrestling
2025_in_professional_wrestling
Irish musician (1971–2018)
described as having "great melodies." Months before she died, O'Riordan tested the resonance and the acoustics of the Glenstal Abbey church in Ireland
Dolores_O'Riordan
16, 2025). "'You, Me & Her' Releases First Trailer; Romantic Comedy Will Test Fithian Group's Digital Distribution Marketplace (Exclusive)". Variety. Retrieved
List of American films of 2025
List_of_American_films_of_2025
messages written on walls and floors by Subject 16, the previous Animus test subject that Lucy and Vidic occasionally mention. Assassin's Creed II continues
List of Assassin's Creed characters
List_of_Assassin's_Creed_characters
1954 Japanese film by Akira Kurosawa
form. In an interview for the Criterion Collection, the director George Lucas cited Seven Samurai as his favorite film by Akira Kurosawa. The Soviet filmmaker
Seven_Samurai
Del Rey, imprint of Random House, Inc. p. 153. ISBN 978-0-345-50129-5. Lucas arrived on June 20, [1983]. "Harrison was in really terrible pain", he says
List of film and television accidents
List_of_film_and_television_accidents
American animated television series
the original on September 9, 2023. Retrieved November 2, 2015. Thomas, Lucas M. (January 11, 2007). "Cartoon Network Racing Review". IGN. News Corporation
Dexter's_Laboratory
Characteristic of a group of organisms with a common ancestor
unique ancestor commonly referred to as the last universal common ancestor (LUCA) of all life on Earth. Common descent is an effect of speciation, in which
Common_descent
with Jordan Fisher in the new "Party Royale" game mode on May 1, 2020, to test out that mode, while a multi-part concert by Dillon Francis, Steve Aoki,
Fortnite_seasonal_events
Video game franchise
Archived from the original on 8 March 2022. Retrieved 4 March 2026. Sullivan, Lucas (21 December 2017). ""I Wasn't Some Hired Gun Who Was Just Brought in to
Horizon_(video_game_series)
2021 video game
non-chronological order. In August 1941, Australian Army demolitionist Private Lucas Riggs (Martin Copping) is stationed at Tobruk as part of the Rats of Tobruk
Call_of_Duty:_Vanguard
Characters from post-apocalyptic drama series
from the original on January 6, 2023. Retrieved January 7, 2023. Manfredi, Lucas (January 6, 2023). "'The Last of Us' Creators Talk the Scrapped Movie Adaptation
Characters of The Last of Us (TV series)
Characters_of_The_Last_of_Us_(TV_series)
Film character played by Wesley Snipes
Blade II, Snipes stated: "I love playing this role. It's fun as an actor to test your skills at doing a sequel, to see if you can recreate something that
Blade (New Line franchise character)
Blade_(New_Line_franchise_character)
Fictional alien race
Zerg art style revealed". PC Gamer. Retrieved June 29, 2026. Sullivan, Lucas (November 8, 2011). "StarCraft II: Heart of the Swarm interview—Blizzard
Zerg
LUCAS PRIMALITY-TEST
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LUCAS PRIMALITY-TEST