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On prime divisors in Fibonacci and Lucas sequences
In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael, states that, for any nondegenerate Lucas sequence of
Carmichael's_theorem
Function in mathematical number theory
primitive root modulo n, which Carmichael sometimes refers to as a primitive φ {\displaystyle \varphi } -root modulo n.) Theorem 2—For every positive integer
Carmichael_function
Theorem on modular exponentiation
{\displaystyle n} must be coprime. The theorem is further generalized by some of Carmichael's theorems. The theorem may be used to easily reduce large powers
Euler's_theorem
A prime p divides a^p–a for any integer a
In number theory, Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p. In
Fermat's_little_theorem
On prime divisors of differences two nth powers
{Z}}(W_{n})} . Carmichael's theorem Wilson prime Kaprekar's constant Fermat's little theorem Palindromic numbers Harshad numbers Dirichlet's theorem on arithmetic
Zsigmondy's_theorem
17th-century conjecture proved by Andrew Wiles in 1994
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that there are no positive integers a
Fermat's_Last_Theorem
Composite number in number theory
Fermat's Little Theorem does not hold. This fact precludes the use of that theorem as an absolute test of primality. The Carmichael numbers form the
Carmichael_number
theorem (number theory) Brauer–Siegel theorem (number theory) Brun's theorem (number theory) Brun–Titchmarsh theorem (number theory) Carmichael's theorem
List_of_theorems
American mathematician (1879–1967)
Fermat's Little Theorem although they are not primes), Carmichael's totient function conjecture, Carmichael's theorem, and the Carmichael function, all
Robert_Daniel_Carmichael
Computation modulo a fixed integer
important theorems relating to modular arithmetic: Carmichael's theorem Chinese remainder theorem Euler's theorem Fermat's little theorem (a special
Modular_arithmetic
Certain constant-recursive integer sequences
primitive. Carmichael's theorem states that all but finitely many of the terms in a Lucas sequence have a primitive prime factor. Indeed, Carmichael (1913)
Lucas_sequence
Prime number in the Fibonacci sequence
Fk, but "at least" one new characteristic prime from Carmichael's theorem. Carmichael's Theorem applies to all Fibonacci numbers except four special cases:
Fibonacci_prime
Positive integer of the form (2^(2^n))+1
2307/2031878, JSTOR 2031878 Yabuta, M. (2001), "A simple proof of Carmichael's theorem on primitive divisors" (PDF), Fibonacci Quarterly, 39 (5): 439–443
Fermat_number
Numbers obtained by adding the two previous ones
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 numbers
Fibonacci_sequence
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
Partial results found before the complete proof
Theorem is a theorem in number theory, originally stated by Pierre de Fermat in 1637 and proven by Andrew Wiles in 1995. The statement of the theorem
Proof of Fermat's Last Theorem for specific exponents
Proof_of_Fermat's_Last_Theorem_for_specific_exponents
Problem in number theory on equal totients
{\displaystyle \varphi (m)=\varphi (n)} . Robert Carmichael first stated this conjecture in 1907, but as a theorem rather than as a conjecture. However, his
Carmichael's totient function conjecture
Carmichael's_totient_function_conjecture
Analytic number theory
In mathematics, the Hardy–Ramanujan theorem, proved by Ramanujan and checked by Hardy states that the normal order of the number ω ( n ) {\displaystyle
Hardy–Ramanujan_theorem
Number divisible only by 1 and itself
than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself
Prime_number
Result on density of prime numbers
and so it is also called the Bertrand–Chebyshev theorem or Chebyshev's theorem. Chebyshev's theorem can also be stated as a relationship with π ( x )
Bertrand's_postulate
Characterization of even perfect numbers
The Euclid–Euler theorem is a theorem in number theory that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and
Euclid–Euler_theorem
Algorithm for public-key cryptography
follows from Euler's theorem. More generally, for any e and d satisfying ed ≡ 1 (mod λ(n)), the same conclusion follows from Carmichael's generalization of
RSA_cryptosystem
Number of integers coprime to and less than n
primes. Carmichael function (λ) Dedekind psi function (𝜓) Divisor function (σ) Duffin–Schaeffer conjecture Generalizations of Fermat's little theorem Highly
Euler's_totient_function
Square of a triangular number
_{k=1}^{n}k\right)^{2}.} This identity is sometimes called Nicomachus's theorem, after Nicomachus of Gerasa (c. 60 – c. 120 CE). Nicomachus, at the end
Squared_triangular_number
Conditions under which the congruence x^3 equals p (mod q) is solvable
Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q)
Cubic_reciprocity
Number theory conjecture
for the second equivalence to hold, since if p is prime, Fermat's little theorem states that a p − 1 ≡ 1 ( mod p ) {\displaystyle a^{p-1}\equiv 1{\pmod
Agoh–Giuga_conjecture
as of September 2022[update]. The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic
List_of_conjectures
Prime such that p^2 divides 2^(p-1)-1
divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes
Wieferich_prime
Pair of integers related by their divisors
Pedersen & te Riele (2003), Sándor & Crstici (2004)]. The Thābit ibn Qurrah theorem is a method for discovering amicable numbers invented in the 9th century
Amicable_numbers
Linear congruence theorem Successive over-relaxation Chinese remainder theorem Fermat's little theorem Proofs of Fermat's little theorem Fermat quotient
List_of_number_theory_topics
Composite number with special property
The special case K1 is the Carmichael numbers. There are infinitely many n-Knödel numbers for a given n. Due to Euler's theorem every composite number m
Knödel_number
Mathematics analytic function
algebraically transcendental are transcendentally transcendental. Hölder's theorem shows that the gamma function is in this category. Hypertranscendental
Hypertranscendental_function
function such as a step function. Carmichael's totient function conjecture was stated as a theorem by Robert Daniel Carmichael in 1907, but in 1922 he pointed
List_of_incomplete_proofs
Cryptographic attack on the RSA system
cipher text C is given by Cd ≡ (Me)d ≡ Med ≡ M (mod N) (using Euler's Theorem). Using the Euclidean algorithm, one can efficiently recover the secret
Wiener's_attack
Result of multiplying four instances of a number together
other fourth powers (the n = 4 case of Fermat's Last Theorem; see Fermat's right triangle theorem). Euler conjectured that a fourth power cannot be written
Fourth_power
Composite number that passes Fermat's probable primality test
important class of pseudoprimes that come from Fermat's little theorem. Fermat's little theorem states that if p {\displaystyle p} is prime and a {\displaystyle
Fermat_pseudoprime
1996 book by Ian Stewart
Interest – Fermat's Last Theorem Chapter 4 – Parallel Thinking – non-Euclidean geometry Chapter 5 – The Miraculous Jar – Cantor's theorem and cardinal numbers
From_Here_to_Infinity_(book)
Integer side lengths of a right triangle
same for the three elements). The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula
Pythagorean_triple
Permutation group that preserves no non-trivial partition
every n > 2. Block (permutation group theory) Jordan's theorem (symmetric group) O'Nan–Scott theorem, a classification of finite primitive groups into various
Primitive_permutation_group
Natural number
different ways, a property that can be connected (via the sum of two squares theorem) to its factorization 5 × 13 × 17 as the product of the three smallest
1105_(number)
Group of units of the ring of integers modulo n
generally ignored and some authors choose not to include the case of n = 1 in theorem statements. Modulo 2 there is only one coprime congruence class, [1], so
Multiplicative group of integers modulo n
Multiplicative_group_of_integers_modulo_n
Odd composite number which passes the given congruence
the above equation which can be deduced from Fermat's little theorem. Fermat's theorem asserts that if p is prime, and coprime to a, then ap−1 ≡ 1 (mod
Euler_pseudoprime
Special type of prime number
prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem is a congruence relation satisfied by all prime numbers greater
Wolstenholme_prime
Belgian mathematician (1866–1962)
a Belgian mathematician. He is best known for proving the prime number theorem. The King of Belgium ennobled him with the title of baron. De La Vallée
Charles-Jean de La Vallée Poussin
Charles-Jean_de_La_Vallée_Poussin
2021) Duffin–Schaeffer theorem (Dimitris Koukoulopoulos, James Maynard, 2019) Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Prime number of the form 2^n – 1
because of their close connection to perfect numbers: the Euclid–Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne
Mersenne_prime
German mathematician (1861–1941)
Deutsche Mathematiker-Vereinigung. Engel group Engel expansion Engel's theorem Carmichael, R. D. (1923). "Volume III of Lie's Memoirs". Bull. Amer. Math. Soc
Friedrich Engel (mathematician)
Friedrich_Engel_(mathematician)
American mathematician (1884–1944)
general relativity. Today, Birkhoff is best remembered for the ergodic theorem. The George D. Birkhoff House, his residence in Cambridge, Massachusetts
George_David_Birkhoff
Integer having a non-trivial divisor
unique up to the order of the factors. This fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine
Composite_number
Four integers where the sum of the squares of three equals the square of the fourth
Lagrange's four-square theorem (every natural number can be represented as the sum of four integer squares) Legendre's three-square theorem (which natural numbers
Pythagorean_quadruple
Integer filtered out using a sieve similar to that of Eratosthenes
with primes, such as asymptotic behaviour according to the prime number theorem; also, a version of Goldbach's conjecture has been extended to them. There
Lucky_number
Probabilistic primality test
odd prime, it passes the test because of two facts: by Fermat's little theorem, a n − 1 ≡ 1 ( mod n ) {\displaystyle a^{n-1}\equiv 1{\pmod {n}}} (this
Miller–Rabin_primality_test
Product of an integer with itself
square number, while other divisors come in pairs. Lagrange's four-square theorem states that any positive integer can be written as the sum of four or fewer
Square_number
Natural number
sequence of "Fermat near misses" defined, in reference to Fermat's Last Theorem, as numbers of the form 1 + z 3 {\displaystyle 1+z^{3}} which are also
1729_(number)
Prime number congruent to 1 mod 4
squares; this characterization is Fermat's theorem on sums of two squares. Equivalently, by the Pythagorean theorem, they are the odd prime numbers p {\displaystyle
Pythagorean_prime
2018 film by J. A. Bayona
earlier films, is no longer inhabited. The website was designed by Chaos Theorem, a creative digital storytelling company founded by Jack Anthony Ewins
Jurassic World: Fallen Kingdom
Jurassic_World:_Fallen_Kingdom
Integers that satisfy a specific condition
rare. Fermat's test for compositeness, which is based on Fermat's little theorem, works as follows: given an integer n, choose some integer a that is not
Probable_prime
Probabilistic primality test
test to determine whether a number is a probable prime. Fermat's little theorem states that if p is prime and a is not divisible by p, then a p − 1 ≡ 1
Fermat_primality_test
Figurate number
{n}{2}}\right)=n^{2}=(T_{n}-T_{n-1})^{2}.} This property, colloquially known as the theorem of Theon of Smyrna, is visually demonstrated in the following sum, which
Triangular_number
Triangle with integer side lengths
incenter and the circumcenter of an integer triangle, given by Euler's theorem as R 2 − 2 R r {\displaystyle R^{2}-2Rr} is rational. A Heronian triangle
Integer_triangle
Computational physics simulation tool
observables written in anti-normal order follow the optical equivalence theorem. This means that it is essentially the density matrix put into normal order
Husimi_Q_representation
Geometry with 7 points and 7 lines
collinear points (on the same line) to collinear points. By the Fundamental theorem of projective geometry, the full collineation group (or automorphism group
Fano_plane
Number equal to the sum of its proper divisors
even perfect numbers are of this form. This is known as the Euclid–Euler theorem. It is not known whether there are any odd perfect numbers, nor whether
Perfect_number
Xueguang; Xu, Jianyu; Wang, Xinyi; Xia, Tony (December 2023). "TheoremQA: A Theorem-driven Question Answering Dataset". In Bouamor, Houda; Pino, Juan;
Language_model_benchmark
American mathematician and astronomer (1871–1948)
Moon is named in his honor. MacMillan, W. D. (1910). "A new proof of the theorem of Weierstrass concerning the factorization of a power series". Bulletin
William_Duncan_MacMillan
Algorithm for determining whether a number is prime
except 1 is divisible by at least one prime number by the Fundamental Theorem of Arithmetic. Therefore the algorithm need only search for prime divisors
Primality_test
Type of prime number
{\displaystyle !} " denotes the factorial function; compare this with Wilson's theorem, which states that every prime p {\displaystyle p} divides ( p − 1 ) !
Wilson_prime
Number that is more than the sum of its proper divisors
Prielipp (1970), Theorem 1, pp. 693–694. Prielipp (1970), Theorem 2, p. 694. Prielipp (1970), Theorem 7, p. 695. Prielipp (1970), Theorem 3, p. 694. Sándor
Deficient_number
Television comedy series
"Machadaynu", performed by Tony Rudd (played by Kevin Eldon) and Antony Carmichael's "The Rapping Song" are beaten in the contest by Toni Baxter's track,
Look_Around_You
Type of feedforward neural network
layers with a stride greater than one ignore the Nyquist–Shannon sampling theorem and might lead to aliasing of the input signal While, in principle, CNNs
Convolutional_neural_network
Concept in modular arithmetic
function, and is denoted as U(n) or U(Zn). As a consequence of Lagrange's theorem, the order of a (mod n) always divides φ(n). If the order of a is actually
Multiplicative_order
Measurement of a quantum system which minimally disturbs it
system on average, but also disturbs the state very little. From Busch's theorem any quantum system is necessarily disturbed by measurement, but the amount
Weak_measurement
Probable prime that is composite
this, there are no pseudoprimes with respect to them. Fermat's little theorem states that if p is prime and a is coprime to p, then ap−1 − 1 is divisible
Pseudoprime
Sporadic simple group
Student Texts, vol. 45, Cambridge University Press, ISBN 978-0-521-65378-7 Carmichael, Robert D. (1956) [1937], Introduction to the theory of groups of finite
Mathieu_group_M12
(1-M_{\infty }^{2})\phi _{xx}+\phi _{yy}+\phi _{zz}=0} (subsonic) From Divergence Theorem ∭ V ( ∇ ⋅ F ) d V = ∬ S F ⋅ n d S {\displaystyle \iiint \limits _{V}\left(\nabla
Aerodynamic potential-flow code
Aerodynamic_potential-flow_code
eponymous donkey. Cantor–Bernstein–Schröder theorem (also known by other variations, such as Schröder-Bernstein theorem) first proved by Richard Dedekind Cantor
List of examples of Stigler's law
List_of_examples_of_Stigler's_law
Two raised to an integer power
q. And since 31 does not divide q and q measures 496, the fundamental theorem of arithmetic implies that q must divide 16 and be among the numbers 1
Power_of_two
Serkis, English actor Millie Gibson, British actress July 2 – Caitlin Carmichael, American actress July 4 – Alex R. Hibbert, American actor July 6 – Dylan
2004_in_film
Product of prime numbers, plus one
after the ancient Greek mathematician Euclid, in connection with Euclid's theorem that there are infinitely many prime numbers. A Euclid number of the second
Euclid_number
Numbers with many divisors
as small as possible, but not too many of the same. By the fundamental theorem of arithmetic, every positive integer n has a unique prime factorization:
Highly_composite_number
Recursive integer sequence
numbers can be interpreted as a special case of the Bertrand's ballot theorem. Specifically, C n {\displaystyle C_{n}} is the number of ways for a candidate
Catalan_number
Class of natural numbers with many divisors
factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Factorization forms Prime Composite Semiprime Pronic Sphenic
Superior highly composite number
Superior_highly_composite_number
Model of quantum optics
to consider A 2 {\displaystyle A^{2}} terms that according to a no-go theorem, may prevent the transition. Both limitations can be circumvented by applying
Dicke_model
with n. In fact, these integers are roots of unity modulo n by Euler's theorem, and the other integers cannot be roots of unity modulo n, because they
Root_of_unity_modulo_n
Integral of the reciprocal of the logarithm, important in the prime number theorem. Exponential integral Trigonometric integral: Including Sine Integral and
List of mathematical functions
List_of_mathematical_functions
Natural number
sum of the cubes of the first nine positive integers, by Nicomachus's theorem), centered octagonal number, lowest number with exactly 15 odd divisors
2000_(number)
Number that is the numerator of the generalized harmonic number H_(n,2)
numbers are named after Joseph Wolstenholme, who proved Wolstenholme's theorem on modular relations of the generalized harmonic numbers. Weisstein, Eric
Wolstenholme_number
Symbols for constants, special functions
\mathrm {H} } represents: the Eta function of Ludwig Boltzmann's H-theorem ("Eta" theorem), in statistical mechanics Information theoretic (Shannon) entropy
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Number used for counting
replaced by its negation. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem. Starting at 0 or 1 has
Natural_number
Product of two prime numbers
distinct ways (23 rows and 73 columns, or 73 rows and 23 columns). Chen's theorem Sphenic number, a product of three distinct primes Parity problem (sieve
Semiprime
Norwegian mathematician (1842–1899)
45 (7): 513–514. doi:10.1090/S0002-9904-1939-07032-8. ISSN 0002-9904. Carmichael, R. D. (1930). "Book Review: vol. IV of Sophus Lie's Gesammelte Abhandlungen
Sophus_Lie
Odd composite number which passes the given congruence
These tests are over twice as strong as tests based on Fermat's little theorem. Every Euler–Jacobi pseudoprime is also a Fermat pseudoprime and an Euler
Euler–Jacobi_pseudoprime
Probabilistic primality test
The idea behind the test was discovered by M. M. Artjuhov in 1967 (see Theorem E in the paper). This test has been largely superseded by the Baillie–PSW
Solovay–Strassen primality test
Solovay–Strassen_primality_test
Type of figurate number
triangular. Centered polygonal number Polyhedral number Fermat polygonal number theorem See Basel problem. Tattersall, James J. (2005). Elementary Number Theory
Polygonal_number
American mathematician
three claimed proofs of a theorem stating that all finite division algebras were commutative, now known as Wedderburn's theorem. The proofs all made clever
Leonard_Eugene_Dickson
Markovian quantum master equation for density matrices (mixed states)
evolution of the expectation values of observables, given by the Ehrenfest theorem. Corresponding to the trace-preserving property of the Schrödinger picture
Lindbladian
Composite number which passes Miller–Rabin primality test
probable prime (PRP). We pick base a = 3 and, inspired by Fermat's little theorem, calculate: 3 31696 ≡ 1 ( mod 31697 ) {\displaystyle 3^{31696}\equiv 1{\pmod
Strong_pseudoprime
American songwriter (born 1940)
Records / 2016 "When Love Is Near" The Original Caste 1969 Tartaglian Theorem / Capitol Records / 2012 "You and Me Against the World (song)" Helen Reddy
Paul_Williams_(songwriter)
Algorithms to generate prime numbers
likely be quickly discovered by causing failed operations, except when a Carmichael number happens to be chosen in the case of RSA. A less common choice is
Generation_of_primes
Figurate number
Euler's theory of integer partitions, as expressed in his pentagonal number theorem. The number of dots inside the outermost pentagon of a pattern forming
Pentagonal_number
CARMICHAELS THEOREM
CARMICHAELS THEOREM
Boy/Male
Scottish Gaelic
Friend of Saint Michael.
Boy/Male
Gaelic
Son of the one who served Saint Michael.
Boy/Male
Australian, Gaelic, Scottish
Follower of Michael; Friend of Saint Michael
CARMICHAELS THEOREM
CARMICHAELS THEOREM
Girl/Female
American, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi
Nectar; Delightful
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Heaven
Girl/Female
Muslim
Beautiful
Girl/Female
Tamil
Nakiska | நாகீஸகாÂ
Girl/Female
Irish
Brave.
Girl/Female
Gujarati, Hindu, Indian, Kannada, Tamil
Successfull Lady; To Succeed
Girl/Female
Hebrew American
Princess.
Girl/Female
Australian, Irish
Sea Child; Born of the Sea
Surname or Lastname
English
English : habitational name possibly from any of three places in Devon called Lincombe, named in Old English with līn ‘flax’ or lind ‘lime tree’ + cumb ‘valley’.
Boy/Male
Muslim
Muhammad Ibn Ismail al-bukha
CARMICHAELS THEOREM
CARMICHAELS THEOREM
CARMICHAELS THEOREM
CARMICHAELS THEOREM
CARMICHAELS THEOREM
n.
One who constructs theorems.
n.
The enunciation of a self-evident problem, in distinction from an axiom, which is the enunciation of a self-evident theorem.
n.
A statement of a principle to be demonstrated.
v. t.
To formulate into a theorem.
n.
A numerical coefficient in any particular case of the binomial theorem.
a.
Theorematic.
a.
Containing many names or terms; multinominal; as, the polynomial theorem.
n.
That which is considered and established as a principle; hence, sometimes, a rule.
a.
Of or pertaining to a theorem or theorems; comprised in a theorem; consisting of theorems.
n.
A theorem or proposition so easy of demonstration as to be almost self-evident.
a.
Alt. of Theorematical