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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 am≡1(modn)
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
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
Function whose domain is the positive integers
λ(n) be Liouville's function. Then |λ(n)|μ(n)=λ(n)|μ(n)|=μ(n), and λ(n)μ(n)=|μ(n)|=μ2(n). Let λ(n) be Carmichael's function. Then λ(n)∣ϕ(n).
Arithmetic_function
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
Arithmetic operation
exponent. For example, if f is a function whose valued can be multiplied, f∘n denotes exponentiation with respect to function composition, whereas fn may denote
Exponentiation
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
Stirling's approximation for n ! {\displaystyle n!} , or via generating functions. The only odd Catalan numbers C n {\displaystyle C_{n}} are those
Catalan_number
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
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
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)
Number divisible only by 1 and itself
which asks where the zeros of the Riemann zeta function ζ(s) are located. This function is an analytic function on the complex numbers. For complex numbers
Prime_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
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
Composite number which passes Miller–Rabin primality test
coprime bases (the Carmichael numbers), there are no composites that are strong pseudoprimes to all bases, i.e. there are no "strong Carmichael numbers". Let
Strong_pseudoprime
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
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
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
Composite number with special property
{\displaystyle \varphi } is Euler's totient function. A 1-Knödel number is commonly known as a Carmichael number, and a 3-Knödel number is also known
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
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)
Numbers with a certain property involving recursive summation
eventually reaches 1 when iterated over the perfect digital invariant function for p=2. The origin of happy numbers is not clear. Happy numbers were brought
Happy_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)
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
Integers occurring in the coefficients of the Taylor series of 1/cosh t
expansion 1cosht=2et+e−t=∑n=0∞Entnn!, where cosh(t) is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials
Euler_number
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
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
Golf tournament in the United States
arose when it was alleged that Paige's teammate Malosi was using his slope function on his rangefinder, which was strictly prohibited for the event. Ryan Whitney
Internet_Invitational
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)
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
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
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
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
Technique to solve differential equations
mathematicians including Charles James Hargreave, George Boole, Bownin, Carmichael, Doukin, Graves, Murphy, William Spottiswoode and Sylvester. Treatises
Operational_calculus
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
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)
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
Natural number
distinct sets 1885 is a Quasi–Carmichael number, and possibly the only number whose preceding number, 1886, is also a Quasi–Carmichael number. 1885 is a Zeisel
1000_(number)
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
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
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
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
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 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
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
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
Triangular array of natural numbers
exactly k blocks, we use the Narayana number N(n,k). The generating function for the Narayana numbers is ∑n=1∞∑k=1nN(n,k)zntk−1=1−z(t+1)−1−2z(t+1)+z2(t−1)22tz
Narayana_number
American mathematician (born 1967)
totient function. In 1998, he published a paper that studied in detail the range of this function and established that Carmichael's totient function conjecture
Kevin_Ford_(mathematician)
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
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
Odd composite number which passes the given congruence
There are no numbers which are Euler–Jacobi pseudoprimes to all bases as Carmichael numbers are. Solovay and Strassen showed that for every composite n, for
Euler–Jacobi_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
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)
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
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
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
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
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 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
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
Type of prime number
{\displaystyle (p-1)!+1} , where " ! {\displaystyle !} " denotes the factorial function; compare this with Wilson's theorem, which states that every prime p {\displaystyle
Wilson_prime
1952 film by Henry Levin
Gilbreth Jeffrey Hunter as Dr. Bob Grayson Edward Arnold as Sam Harper Hoagy Carmichael as Thomas George Bracken Barbara Bates as Ernestine Gilbreth Robert Arthur
Belles_on_Their_Toes_(film)
Integer with a specific property
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Sierpiński_number
Number in the 5th cell of any row of Pascal's triangle
natural number. In that case x is the nth pentatope number. The generating function for pentatope numbers is x ( 1 − x ) 5 = x + 5 x 2 + 15 x 3 + 35 x 4 +
Pentatope_number
Concept in combinatorics
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Cake_number
Three-dimensional figurate centered icosahedral numbers
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Centered_icosahedral_number
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
Product of two distinct primes ≡ 3 (mod 4)
principal square root of a modulo n The function f : Qn → Qn defined by f(x) = x2 mod n is a permutation. The inverse function of f is: f−1(x) = x((p−1)(q−1)+4)/8
Blum_integer
county of the city of Belfast, in Ireland, journalist, with Arabella Carmichael Osborne his now wife, and to enable him to marry again, and for other
List of acts of the Parliament of the United Kingdom from 1920
List_of_acts_of_the_Parliament_of_the_United_Kingdom_from_1920
Number that is not in the range of Euler's totient function
which is not a totient number: it is not in the image of Euler's totient function φ, that is, the equation φ(x) = n has no solution x. In other words, n
Nontotient
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 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
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)
Former 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
Positive integer that is an integer power of another positive integer
(k))\approx 0.874464368\dots } where μ(k) is the Möbius function and ζ(k) is the Riemann zeta function. According to Euler, Goldbach showed (in a now-lost
Perfect_power
Combinatorial sequence of numbers
have been found only for n ≤ 9 {\displaystyle n\leq 9} . A Boolean function is a function that takes as input n Boolean variables (that is, values that can
Dedekind_number
Number of ways to pair up n objects
is the value at zero of the n-th derivative of this function. The exponential generating function can be derived in a number of ways; for example, taking
Telephone number (mathematics)
Telephone_number_(mathematics)
Mathematical formula
pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant
Multiplicative_digital_root
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
travel, tourism, insurance
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