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Function in mathematical number theory
In number theory, a branch of mathematics, the Carmichael function λ(n) of a positive integer n is the smallest positive integer m such that a m ≡ 1 (
Carmichael_function
Problem in number theory on equal totients
In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ ( n ) {\displaystyle \varphi
Carmichael's totient function conjecture
Carmichael's_totient_function_conjecture
Number of integers coprime to and less than n
the product of the first 120569 primes. Carmichael function (λ) Dedekind psi function (𝜓) Divisor function (σ) Duffin–Schaeffer conjecture Generalizations
Euler's_totient_function
Function whose domain is the positive integers
a_{k}\;\land \;n=a_{1}+a_{2}+\cdots +a_{k}\right\}\right|.} λ(n), the Carmichael function, is the smallest positive number such that a λ ( n ) ≡ 1 ( mod n
Arithmetic_function
Mangoldt function, Λ(n) = log p if n is a positive power of the prime p Carmichael function: λ ( n ) = {\displaystyle \lambda (n)=} The smallest integer m {\displaystyle
List of mathematical functions
List_of_mathematical_functions
Trinbagonian-American activist (1941–1998)
(/ˈkwɑːmeɪ ˈtʊəreɪ/ KWAH-may TOOR-ay; born Stokely Standiford Churchill Carmichael; June 29, 1941 – November 15, 1998) was a Trinidadian and American activist
Stokely_Carmichael
Topics referred to by the same term
function, λ(τ), a highly symmetric holomorphic function on the complex upper half-plane Carmichael function, λ(n), in number theory and group theory Lambda
Lambda_function
Composite number in number theory
In number theory, a Carmichael number is a composite number n {\displaystyle n} which in modular arithmetic satisfies the congruence relation: b n
Carmichael_number
American mathematician (1879–1967)
although they are not primes), Carmichael's totient function conjecture, Carmichael's theorem, and the Carmichael function, all significant in number theory
Robert_Daniel_Carmichael
and φ {\displaystyle \varphi } are respectively the Carmichael function and Euler's totient function.[clarification needed] A root of unity modulo n is
Root_of_unity_modulo_n
Decimal representation of a number whose digits are periodic
factor of λ(49) = 42, where λ(n) is known as the Carmichael function. This follows from Carmichael's theorem which states that if n is a positive integer
Repeating_decimal
Pseudorandom number generator
}}(M)}\right){\bmod {M}}} , where λ {\displaystyle \lambda } is the Carmichael function. (Here we have λ ( M ) = λ ( p ⋅ q ) = lcm ( p − 1 , q − 1 ) {\displaystyle
Blum_Blum_Shub
Concept in modular arithmetic
generates it. The order of a (mod n) also divides λ(n), a value of the Carmichael function, which is an even stronger statement than the divisibility of φ(n)
Multiplicative_order
A prime p divides a^p–a for any integer a
and q of n. Fermat's little theorem is also related to the Carmichael function and Carmichael's theorem, as well as to Lagrange's theorem in group theory
Fermat's_little_theorem
Natural number
one way. 224 is the smallest k with λ(k) = 24, where λ(k) is the Carmichael function. The mathematician and philosopher Alex Bellos suggested in 2014
224_(number)
Public-key cryptosystem
\lambda (n))=1} , where λ ( n ) {\displaystyle \lambda (n)} is the Carmichael function. Compute d := e − 1 mod λ ( n ) {\displaystyle d:=e^{-1}{\bmod {\lambda
Key_encapsulation_mechanism
Symbols for constants, special functions
density ecliptic longitude in astronomy the Liouville function in number theory the Carmichael function in number theory the empty string in formal grammar
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Cryptographic attack on the RSA system
≡ 1 (mod λ(N)), where λ(N) denotes the Carmichael function, though sometimes φ(N), the Euler's totient function, is used (note: this is the order of the
Wiener's_attack
Modular arithmetic concept
no primitive roots modulo 15. Indeed, λ(15) = 4, where λ is the Carmichael function. (sequence A002322 in the OEIS) Numbers n {\displaystyle n} that
Primitive_root_modulo_n
New Zealand theoretical physicist
Howard John Carmichael (born 17 January 1950) is a British-born New Zealand theoretical physicist specialising in quantum optics and the theory of open
Howard_Carmichael
Group of units of the ring of integers modulo n
common multiple of the orders in the cyclic groups, is given by the Carmichael function λ ( n ) {\displaystyle \lambda (n)} (sequence A002322 in the OEIS)
Multiplicative group of integers modulo n
Multiplicative_group_of_integers_modulo_n
1999 United States Supreme Court case
Kumho Tire Co. v. Carmichael, 526 U.S. 137 (1999), is a United States Supreme Court case that applied the Daubert standard to expert testimony from non-scientists
Kumho_Tire_Co._v._Carmichael
Computational simulation method for open quantum systems
known as Quantum Trajectory Theory developed by Carmichael. Other contemporaneous works on wave-function-based Monte Carlo approaches to open quantum systems
Quantum_jump_method
Numbers that contain only the digit 1
because p is prime. Therefore, unless q divides b − 1, p divides the Carmichael function of q, which is even and equal to q − 1. Any positive multiple of
Repunit
Number used for counting
a list of objects in a specific order. More precisely, a sequence is a function that assigns an object to each position in that list. The positions themselves
Natural_number
Class of natural numbers with many divisors
{d(n)}{n^{\varepsilon }}}\geq {\frac {d(k)}{k^{\varepsilon }}}} where d(n), the divisor function, denotes the number of divisors of n. The term was coined by Ramanujan
Superior highly composite number
Superior_highly_composite_number
Natural number
1007/978-0-387-21850-2. ISBN 0-387-95332-9. MR 1866957. Carmichael, R. D. (1910). "Note on a new number theory function". Bulletin of the American Mathematical Society
1105_(number)
Integer having a non-trivial divisor
Sieve of Eratosthenes Table of prime factors Divisor function Prime omega function Möbius function Pettofrezzo & Byrkit 1970, pp. 23–24. Long 1972, p. 16
Composite_number
Mathematics analytic function
Transcendental Functions", Mathematische Annalen 48:1-2:49-74 (1896) doi:10.1007/BF01446334 R. D. Carmichael, "On Transcendentally Transcendental Functions", Transactions
Hypertranscendental_function
Iterative algorithm on numbers
sequence. Repeat step 2. The sequence is called a Kaprekar sequence and the function K b ( n ) = α − β {\displaystyle K_{b}(n)=\alpha -\beta } is the Kaprekar
Kaprekar's_routine
Number divisible only by 1 and itself
the zeros of the Riemann zeta function ζ ( s ) {\displaystyle \zeta (s)} are located. This function is an analytic function on the complex numbers. For
Prime_number
Handelsgesetze des Erdballs. Vol. 8. 1908. p. 13. Epple & Assefa 2020, p. 146. Carmichael 2001, p. 215. Abiad 2008, p. 144. Abiad, Nisrine (2008). Sharia, Muslim
Sharia_court
Type of positive composite integer
In mathematics, a Lucas–Carmichael number is a positive composite integer n such that If p is a prime factor of n, then p + 1 is a factor of n + 1; n is
Lucas–Carmichael_number
Mathematical function
In mathematics, Hooley's delta function ( Δ ( n ) {\displaystyle \Delta (n)} ), also called Erdős--Hooley delta-function, defines the maximum number of
Hooley's_delta_function
Integer filtered out using a sieve similar to that of Eratosthenes
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Lucky_number
Legendary Indigenous North American creature
'what man has the power to lift those great stones?'". Carmichael concluded that "the exact function of the Wanipigow Thunderbird Nests remains enigmatic"
Thunderbird_(mythology)
Formulation of quantum mechanics
Howard Carmichael in the early 1990s around the same time as the similar formulation, known as the quantum jump method or Monte Carlo wave function (MCWF)
Quantum_Trajectory_Theory
Numbers with a certain property involving recursive summation
eventually reaches 1 when iterated over the perfect digital invariant function for p = 2 {\displaystyle p=2} . The origin of happy numbers is not clear
Happy_number
Recursive integer sequence
binomial coefficients, by Stirling's approximation for n!, or via generating functions. The only Catalan numbers Cn that are odd are those for which n = 2k −
Catalan_number
1993 American film
Cousins, Barney's wife George Hearn as Arthur Bernard Lynn Hamilton as Miss Carmichael Principal photography began on April 6, 1992. Initial filming took place
The_Vanishing_(1993_film)
Triangular array of natural numbers
N ( n , k ) {\displaystyle \operatorname {N} (n,k)} . The generating function for the Narayana numbers is ∑ n = 1 ∞ ∑ k = 1 n N ( n , k ) z n t k −
Narayana_number
Product of two prime numbers
where π ( x ) {\displaystyle \pi (x)} is the prime-counting function and p k {\displaystyle p_{k}} denotes the kth prime. To see this, take
Semiprime
Ten raised to an integer power
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Power_of_10
Figurate number
with the factorial function, a product whose factors are the integers from 1 to n, Donald Knuth proposed the name Termial function, with the notation
Triangular_number
Numbers k where x - phi(x) = k has many solutions
{\displaystyle k} and above 1. Here, ϕ {\displaystyle \phi } is Euler's totient function. There are infinitely many solutions to the equation for k {\displaystyle
Highly_cototient_number
Numbers obtained by adding the two previous ones
a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem). As a result, 8 and 144 (F6 and F12) are the only Fibonacci
Fibonacci_sequence
Arithmetic operation
function xy has no limit at the point (0, 0). One may consider at what points this function does have a limit. More precisely, consider the function f
Exponentiation
Infinite integer series where the next number is the sum of the two preceding it
− 4 ( 18 ) + 6 {\displaystyle 256=322-4(18)+6} The ordinary generating function of the sequence of Lucas numbers is the power series Φ ( x ) = ∑ k = 0
Lucas_number
American comedian and actress (born 1979)
drama, Haddish gained prominence for her roles in the NBC sitcom The Carmichael Show (2015–2017), the TBS series The Last O.G. (2018–2020), the Hulu series
Tiffany_Haddish
American cyclist (born 1960)
Chris Carmichael (born October 24, 1960, in Miami, Florida, United States) is a retired professional cyclist and cycling, triathlon and endurance sports
Chris_Carmichael_(cyclist)
Integer having only small prime factors
algorithms, for example: the general number field sieve), the VSH hash function is another example of a constructive use of smoothness to obtain a provably
Smooth_number
Integers occurring in the coefficients of the Taylor series of 1/cosh t
Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics, specifically
Euler_numbers
Natural number
163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211 2015 – Lucas–Carmichael number 2016 – second-smallest Erdős–Nicolas number, triangular number
2000_(number)
Retrieved September 20, 2022. Fleming, Mike Jr. (December 8, 2022). "Jerrod Carmichael To Host 80th Annual Golden Globe Awards". Deadline Hollywood. Archived
2023_in_American_television
Number raised to the third power
n × n × n. The cube function is the function x ↦ x3 (often denoted y = x3) that maps a number to its cube. It is an odd function, as (−n)3 = −(n3). The
Cube_(algebra)
Number that when multiplied by another number moves its last digit to its front
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Parasitic_number
Type of composite number with an even number of digits
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Vampire_number
Number that remains the same when its digits are reversed
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Palindromic_number
Type of Poulet number
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Super-Poulet_number
Number n where n and totient(n) are coprime
Monthly. 107 (7): 631–634. doi:10.2307/2589118. Retrieved 21 May 2021. Carmichael Multiples of Odd Cyclic Numbers See T. Szele, Über die endlichen Ordnungszahlen
Cyclic_number_(group_theory)
Count of the possible partitions of a set
exponential function and the nonemptiness constraint ≥1 into subtraction by one. An alternative method for deriving the same generating function uses the
Bell_number
minor roles in episodes before eventually coming to the forefront." Les Carmichael, portrayed by Stacy J Gough, was Matty Barton's (Ash Palmisciano) cellmate
List of Emmerdale characters introduced in 2024
List_of_Emmerdale_characters_introduced_in_2024
Computational physics simulation tool
Studies in Modern Optics. ISBN 0521497302 , ISBN 978-0521497305. H. J. Carmichael (2002). Statistical Methods in Quantum Optics I: Master Equations and
Husimi_Q_representation
Technique to solve differential equations
mathematicians including Charles James Hargreave, George Boole, Bownin, Carmichael, Doukin, Graves, Murphy, William Spottiswoode and Sylvester. Treatises
Operational_calculus
Danish mathematician (1885–1981)
Greenland". Geological Survey of Denmark. Retrieved 26 September 2019. Carmichael, R. D. (1925). "Nörlund on Calculus of Differences". Bull. Amer. Math
Niels_Erik_Nørlund
2026 wedding in New York City, U.S.
Noah Baumbach and Greta Gerwig, filmmakers Kate Berlant, comedian Jerrod Carmichael, comedian Josh Charles, actor Jessica Chastain, actress Sacha Baron Cohen
Wedding of Taylor Swift and Travis Kelce
Wedding_of_Taylor_Swift_and_Travis_Kelce
Natural number
(Reduced totient function psi(n): least k such that x^k congruent to 1 (mod n) for all x prime to n; also known as the Carmichael lambda function (exponent of
34_(number)
Type of number introduced by Mike Keith
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Keith_number
Two raised to an integer power
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Power_of_two
Number that is the result of operation on its own digits
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Friedman_number
Number equal to the sum of its proper divisors
_{1}(n)=2n} where σ 1 {\displaystyle \sigma _{1}} is the sum-of-divisors function. This definition is ancient, appearing as early as Euclid's Elements (Book
Perfect_number
Positive integer that is the product of three distinct prime numbers
definition squarefree, because the prime factors must be distinct. The Möbius function of any sphenic number is −1. The cyclotomic polynomials Φ n ( x ) {\displaystyle
Sphenic_number
Type of composite integer
{\displaystyle n} be a natural number. For base b > 1 {\displaystyle b>1} , let the function F b ( n ) {\displaystyle F_{b}(n)} be the digit sum of n {\displaystyle
Smith_number
Product of an integer with itself
Integer that is a perfect square modulo some integer Quadratic function – Polynomial function of degree two Square triangular number – Integer that is both
Square_number
Repeated sum of a number's digits
which allows it to be used as a divisibility rule. The formula for the function d r b : N → ⋃ k = 0 b − 1 { k } , b ∈ N ⩾ 2 {\displaystyle \mathrm {dr}
Digital_root
Numbers whose prime factors all divide the number more than once
9435964368\ldots ,} where p runs over all primes, ζ(s) denotes the Riemann zeta function, and ζ(3) is Apéry's constant. (sequence A082695 in the OEIS) More generally
Powerful_number
Numbers parameterizing ways to partition a set
no two consecutive 1s. For a fixed integer n, the ordinary generating function for Stirling numbers of the second kind { n 0 } , { n 1 } , … {\displaystyle
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
Three raised to an integer power
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Power_of_three
"Stokely Carmichael - Civil Rights Movement, SNCC & Speech". HISTORY. 2019-06-10. Retrieved 2023-12-09. Goldman, John J. (1998-11-16). "Stokely Carmichael, Black
List of people with prostate cancer
List_of_people_with_prostate_cancer
Integer whose multiples are digit rotations
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Cyclic_number
Number that represents a hexagon with a dot in the center
calculate the generating function F ( x ) = ∑ n ≥ 0 H ( n ) x n {\displaystyle F(x)=\sum _{n\geq 0}H(n)x^{n}} . The generating function satisfies F ( x ) =
Centered_hexagonal_number
Turkish Empire (c. 1299–1922)
Archived from the original on 14 January 2023. Retrieved 20 June 2015. Carmichael, Cathie (2012). Ethnic Cleansing in the Balkans: Nationalism and the Destruction
Ottoman_Empire
Human-caused changes to climate on Earth
ISSN 0012-8252. Archived from the original on 16 February 2024. Ramanathan & Carmichael 2008 RIVM 2016. Sand et al. 2015 "IMO 2020 – cutting sulphur oxide emissions"
Climate_change
Means by which a person dies by suicide
Archived from the original on 18 October 2019. Retrieved 5 September 2020. Carmichael V, Whitley R (9 May 2019). "Media coverage of Robin Williams' suicide
Suicide_methods
Condition in which damaged skeletal muscle breaks down rapidly
Safety and Health. 3 December 2021. Retrieved 15 December 2022. Kodadek L, Carmichael Ii SP, Seshadri A, Pathak A, Hoth J, Appelbaum R, Michetti CP, Gonzalez
Rhabdomyolysis
Unsolved problem in mathematics
least seven distinct primes (i.e. ω(n) ≥ 7). Such a number must also be a Carmichael number. In 1980, Cohen and Hagis proved that, for any solution n to the
Lehmer's_totient_problem
2011 film based on John le Carré's novel
Carré as Christmas party guest Katrina Vasilieva as Ann Smiley Laura Carmichael as Sal Phillip Martin Brown as Tufty Thesinger William Haddock as Bill
Tinker Tailor Soldier Spy (film)
Tinker_Tailor_Soldier_Spy_(film)
Composite number which passes Miller–Rabin primality test
which there exist numbers that are pseudoprimes to all coprime bases (the Carmichael numbers), there are no composites that are strong pseudoprimes to all
Strong_pseudoprime
American stand-up comedian
comedians, including Trevor Wallace, Andrew Schulz, Trevor Noah, Jerrod Carmichael, Hasan Minhaj, Theo Von, Tim Dillon, Stavros Halkias, Max Amini, Marcella
Martin_Amini
Integer whose representation contains every digit in its number base
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Pandigital_number
Numbers whose sum of divisors is twice the number plus 1
natural number n for which the sum of all its divisors (the sum-of-divisors function σ ( n ) {\displaystyle \sigma (n)} ) is equal to 2 n + 1 {\displaystyle
Quasiperfect_number
Capital city of Canada
Archived from the original on 4 May 2016. Retrieved 15 November 2015. Carmichael, Kevin (11 June 2011). "From a Burst Bubble, a New Brand of Manufacturing
Ottawa
Algorithm for public-key cryptography
released as part of the public key. Compute λ(n), where λ is Carmichael's totient function. Since n = pq, λ(n) = lcm(λ(p), λ(q)), and since p and q are
RSA_cryptosystem
with constant second difference. Carmichael's totient function conjecture: do all values of Euler's totient function have multiplicity greater than 1
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Display of work at stages in a process
C4Media, Publisher of InfoQ.com, USA 2010, p. 31. Anderson, David J.; Carmichael, Andy (2016). Essential Kanban Condensed. Seattle, WA: Lean Kanban University
Kanban_board
Number of paths between grid corners, allowing diagonal steps
This recurrence relation also leads directly to the generating function ∑ m , n = 0 ∞ D ( m , n ) x m y n = ( 1 − x − y − x y ) − 1 . {\displaystyle
Delannoy_number
Composite number with special property
set of all n-Knödel numbers is denoted Kn. The special case K1 is the Carmichael numbers. There are infinitely many n-Knödel numbers for a given n. Due
Knödel_number
Count of permutations by cycles
, v ) {\displaystyle \zeta (k,v)} are the Riemann zeta function and the Hurwitz zeta function respectively, and even evaluate this integral ∫ 0 1 log
Stirling numbers of the first kind
Stirling_numbers_of_the_first_kind
American YouTuber (born 1986)
September 16, 1987, in Sacramento, California, and raised in the suburb of Carmichael, California. His father is of Filipino descent and his mother is German
Anthony_Padilla
Family of basal chelicerates
Biology. 11 (1): 37–40. Bibcode:2010AquaB..11...37C. doi:10.3354/ab00289. Carmichael, Ruth H.; Brush, Erik (March 2012). "Three decades of horseshoe crab rearing:
Horseshoe_crab
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