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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
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
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
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
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
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-American activist
Stokely_Carmichael
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
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
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
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
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
Natural number
one way. 224 is the smallest k with λ(k) = 24, where λ(k) is the Carmichael function. 224 is the best number according to Bruno Milašus's mathematics
224_(number)
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
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
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
{\displaystyle \lambda (m)} , where λ {\displaystyle \lambda } is the Carmichael lambda function. If we require a coprime to m and only consider n dividing λ (
Power_residue_symbol
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
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
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
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
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
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
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
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
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
Product of the first "n" prime numbers
# {\displaystyle p_{n}\#} ", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying
Primorial
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
Arithmetic operation
example, if f is a function whose valued can be multiplied, f ∘ n {\displaystyle f^{\circ n}} denotes exponentiation with respect to function composition, whereas
Exponentiation
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
Natural number
+ 181 + 191 + 193 + 197 + 199 + 211 2015 = 5 × 13 × 31. It is a Lucas–Carmichael number. 2016 = 25 × 32 × 7. It is the second-smallest Erdős–Nicolas numberand
2000_(number)
Judicial institution in Islam
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
1999 United States Supreme Court case
Kumho Tire Co. v. Carmichael, 526 U.S. 137 (1999), is a United States Supreme Court case which held that the Daubert standard applies to expert testimony
Kumho_Tire_Co._v._Carmichael
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
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
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)
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
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)
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
Mathematical function
In mathematics, Hooley's delta function, also called Erdős--Hooley delta-function, defines the maximum number of divisors of n {\displaystyle n} in [ u
Hooley's_delta_function
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
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
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 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
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)
Probable prime that is composite
Fermat pseudoprime to all values of a that are coprime to x is called a Carmichael number. Catalan pseudoprime Elliptic pseudoprime Euler pseudoprime Euler–Jacobi
Pseudoprime
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
Concept in combinatorics
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Cake_number
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
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)
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 number
many Giuga numbers? Is there a composite Giuga number that is also a Carmichael number? More unsolved problems in mathematics All known Giuga numbers
Giuga_number
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
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
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
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
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
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
Technique to solve differential equations
mathematicians including Charles James Hargreave, George Boole, Bownin, Carmichael, Doukin, Graves, Murphy, William Spottiswoode and Sylvester. Treatises
Operational_calculus
Number that has a perfect number of factors adding up to another perfect number
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Sublime_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
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
American singer, songwriter and actress (born 1986)
to be "part ordinary person, part extraterrestrial celebrity empress functions at the highest level". Stephanie Zacharek of Time felt she gave a "knockout
Lady_Gaga
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
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
American political commentator (born 1991)
people with a sense of community, acceptance, place, and, in line with the function of the organic intellectual, with counter-hegemonic narratives. More specifically
Hasan_Piker
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 Stephen Colbert
Wedding of Taylor Swift and Travis Kelce
Wedding_of_Taylor_Swift_and_Travis_Kelce
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)
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
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)
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
Danish mathematician (1885–1981)
Differenzenrechnung (1924, reprinted 1954) was the first book on complex function solutions of difference equations. His doctoral students include Georg
Niels_Erik_Nørlund
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
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_number
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
Number with a half-integer abundancy index
σ(n)/n = k/2 for an odd integer k, where σ(n) is the sum-of-divisors function, the sum of all positive divisors of n. The first few hemiperfect numbers
Hemiperfect_number
Concatenation of the first n prime numbers
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Smarandache–Wellin_number
Type of natural number
{\sigma (k)}{k^{1+\varepsilon }}}} where σ denotes the sum-of-divisors function. That is, n {\displaystyle n} attains the maximum value of σ ( k ) k 1
Colossally_abundant_number
Type of number in combinatorial mathematics and statistics
"Densities of the Raney distributions" paper, let the ordinary generating function with respect to the index m be defined as follows: B p , r ( z ) := ∑ m
Fuss–Catalan_number
Sum of a number's digits
Encyclopedia of Integer Sequences. Borwein & Borwein (1992) use the generating function of this integer sequence (and of the analogous sequence for binary digit
Digit_sum
British scholar, author, and women's rights advocate (1840–1929)
Charlotte Brown Carmichael Stopes (née Carmichael; 5 February 1840 – 6 February 1929), also known as C. C. Stopes, was a British scholar, author, and
Charlotte_Carmichael_Stopes
Result of multiplying four instances of a number together
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Fourth_power
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
Centered figurate number that represents a decagon with a dot in the center
Centered decagonal number iff 20N + 5 is a Square number. The generating function of the centered decagonal number is x ∗ ( 1 + 8 x + x 2 ) ( 1 − x ) 3 {\displaystyle
Centered_decagonal_number
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)
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
Central part of the adrenal gland
substitute. Adrenal gland Chromaffin cell History of catecholamine research Carmichael, Stephen W. (1997). "8. The Adrenal Medulla". In Bittar, E. Edward; Bittar
Adrenal_medulla
Integer divisible by the number of its divisors
investigated the topic. Wikifunctions has a refactorable number checking function. Divisor function J. Zelinsky, "Tau Numbers: A Partial Proof of a Conjecture and
Refactorable_number
Natural number
unit of length. 62,208 = 3-smooth number 62,210 = Markov number 62,745 = Carmichael number 63,020 = amicable number with 76084 63,261 = number of partitions
60,000
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
Counts pieces of a disk cut by lines
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Lazy_caterer's_sequence
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
Numbers in a type of Lucas sequence
3 . {\displaystyle J_{n}={\frac {2^{n}-(-1)^{n}}{3}}.} The generating function for the Jacobsthal numbers is x ( 1 + x ) ( 1 − 2 x ) . {\displaystyle
Jacobsthal_number
Property of a number
persistence grows tetrationally. Some functions only allow persistence up to a certain degree. For example, the function which takes the minimal digit only
Persistence_of_a_number
Integer named after Reo Fortune
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Fortunate_number
Odd number with specific properties
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Sierpiński_number
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
travel, tourism, insurance
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