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Field of mathematics
Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex
Arithmetic_dynamics
Branch of algebraic geometry
cases of the weight-monodromy conjecture. Anabelian geometry Arithmetic dynamics Arithmetic of abelian varieties Birch and Swinnerton-Dyer conjecture Category
Arithmetic_geometry
Area of mathematics
Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the
Dynamical_systems_theory
Natural number
1088/0026-1394/31/6/013. Peano, Giuseppe (1889). Arithmetices principia, nova methodo exposita [The principles of arithmetic, presented by a new method]. An excerpt
1
Chinese-American mathematician (born 1962)
curves (Yuan, Zhang & W. Zhang 2009 Yuan, Zhang & W. Zhang 2013). In arithmetic dynamics, Zhang (1995a, 2006) posed conjectures on the Zariski density of
Shou-Wu_Zhang
factors Formula for primes Factorization RSA number Fundamental theorem of arithmetic Square-free Square-free integer Square-free polynomial Square number Power
List_of_number_theory_topics
Branch of mathematics
mapping from some algebraic variety to itself. The related theory of arithmetic dynamics studies iteration over the rational numbers or the p-adic numbers
Complex_dynamics
Mathematical model of the time dependence of a point in space
"time" lattice.[clarification needed] Symbolic dynamics Finite state automata Turing machines Arithmetic dynamics Graph dynamical system By its discrete nature
Dynamical_system
associated with arithmetic operations such as addition, subtraction, multiplication and division. Arithmetic dynamics Arithmetic dynamics is the study of
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Venezuelan-American mathematician
Foundation from 2021-2026. Her research interests include arithmetic geometry and arithmetic dynamics in number theory. Salerno was born in Caracas in 1979
Adriana_Salerno
Number
consequently dividing by 0 is generally considered to be undefined in arithmetic. As a numerical digit, 0 plays a crucial role in decimal notation: it
0
Geometrical GCD and LCM algorithm
In recreational mathematics, arithmetic billiards provide a geometrical method to determine the least common multiple (LCM) and the greatest common divisor
Arithmetic_billiards
Branch of pure mathematics
branch of mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties
Number_theory
Modeling a dynamical system's states as infinite sequences of symbols
and dynamical systems Shift space Shift of finite type Complex dynamics Arithmetic dynamics Hadamard, J. (1898). "Les surfaces à courbures opposées et leurs
Symbolic_dynamics
American mathematician (born 1955)
professor of mathematics at Brown University working in arithmetic geometry, arithmetic dynamics, and cryptography. Joseph Silverman received an Sc.B. from
Joseph_H._Silverman
Open problem on 3x+1 and x/2 functions
related to Collatz conjecture. 3x + 1 semigroup Arithmetic dynamics Juggler sequence Modular arithmetic Residue-class-wise affine group It is also known
Collatz_conjecture
Bogomolov conjecture have become a central area of research in both arithmetic dynamics and diophantine geometry, motivating many developments in both fields
Bogomolov_conjecture
Sum of a number's digits
their digit sums with the digit sums of their prime factorizations. Arithmetic dynamics Casting out nines Checksum Digital root Hamming weight Harshad number
Digit_sum
Iterative algorithm on numbers
_{i=0}^{n}b^{i}\right)+k\\&=m\\\end{aligned}}} Mathematics portal Arithmetic dynamics Collatz conjecture Dudeney number Factorion Happy number Kaprekar
Kaprekar's_routine
Number whose square ends in the same digits
digits + 1): print(hensels_lemma(automorphic_polynomial, base, i)) Arithmetic dynamics Kaprekar number p-adic number p-adic analysis Zero-divisor See Gérard
Automorphic_number
Integer sequence in number theory
maximum value at a60 with 972,463 digits, before reaching 1 at a157. Arithmetic dynamics Collatz conjecture Recurrence relation Pickover, Clifford A. (1992)
Juggler_sequence
Numbers with a certain property involving recursive summation
episode 42, a sequence of happy primes is the password to open a door. Arithmetic dynamics Fortunate number Harshad number Lucky number Perfect digital invariant
Happy_number
Type of number introduced by Mike Keith
+ i] sequence.append(n) return sequence[len(sequence) - 1] == x Arithmetic dynamics Fibonacci number Linear recurrence relation Keith, Mike (1987). "Repfigit
Keith_number
Number that cannot be written as an aliquot sum
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Untouchable_number
Repeated sum of a number's digits
visual novel adventure game Nine Hours, Nine Persons, Nine Doors. Arithmetic dynamics Base 9 Casting out nines Digit sum Divisibility rule Hamming weight
Digital_root
Natural number
6174 (six thousand, one hundred [and] seventy-four) is the natural number following 6173 and preceding 6175. It is a Kaprekar's Constant 6174 is a 7-smooth
6174
Number that is less than the sum of its proper divisors
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Abundant_number
Concept in number theory
use of a signed-digit representation to represent each integer. Arithmetic dynamics Dudeney number Factorion Happy number Kaprekar's constant Kaprekar
Narcissistic_number
Area of mathematics
these analogies, coining the term arithmetic topology for this area of study. Arithmetic geometry Arithmetic dynamics Topological quantum field theory
Arithmetic_topology
Base-dependent property of integers
use of a signed-digit representation to represent each integer. Arithmetic dynamics Automorphic number Dudeney number Factorion Happy number Kaprekar's
Kaprekar_number
Special semigroup of positive rational numbers
and multiplicative semigroups", Geometry, Spectral Theory, Groups and Dynamics: Proceedings in Memor y of Robert Brooks. Springer. Ana Caraiani. "Multiplicative
3x_+_1_semigroup
Type of natural number
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Self_number
Mathematical arithmetic dynamics function
In arithmetic dynamics, an arboreal Galois representation is a continuous group homomorphism between the absolute Galois group of a field and the automorphism
Arboreal Galois representation
Arboreal_Galois_representation
Number whose divisors add to a multiple of that number
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Multiply_perfect_number
Mathematics of varieties with integer coordinates
these equations. Diophantine geometry is part of the broader field of arithmetic geometry. Four theorems of fundamental importance in Diophantine geometry
Diophantine_geometry
Sum of all proper divisors of a natural number
In number theory, the aliquot sum s(n) of a positive integer n is the sum of all proper divisors of n, that is, all divisors of n other than n itself.
Aliquot_sum
Number, non-palindrome after repeated sum with reverse
use of a signed-digit representation to represent each integer. Arithmetic dynamics Palindromic number O'Bryant, Kevin (26 December 2012). "Reply to
Lychrel_number
Number that is more than the sum of its proper divisors
numbers into deficient, perfect, or abundant, in his Introduction to Arithmetic (circa 100 CE). However, he applied this classification only to the even
Deficient_number
Mathematical recursive sequence
numbers and cycles of length two that represent amicable pairs. Arithmetic dynamics Weisstein, Eric W. "Aliquot Sequence". MathWorld. Sloane, N. J. A
Aliquot_sequence
Sequence in number theory
not in cycle: cycle.append(x) x = dudeneyf(x, p, b) return cycle Arithmetic dynamics Factorion Happy number Kaprekar's constant Kaprekar number Meertens
Dudeney_number
Theory in number theory
geometry is a theory in arithmetic geometry which describes the way in which the algebraic fundamental group of a certain arithmetic variety X, or some related
Anabelian_geometry
theory Arithmetic topology Arithmetic dynamics Arithmetic geometry at the nLab Sutherland, Andrew V. (September 5, 2013). "Introduction to Arithmetic Geometry"
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Pair of integers related by their divisors
Rashed, Roshdi (1994). The development of Arabic mathematics: between arithmetic and algebra. Vol. 156. Dordrecht, Boston, London: Kluwer Academic Publishers
Amicable_numbers
Natural number
196 (one hundred [and] ninety-six) is the natural number following 195 and preceding 197. 196 is a square number, the square of 14. As the square of a
196_(number)
American mathematician (born 1974)
Mathematics for her contributions to complex dynamics, potential theory, and the emerging field of arithmetic dynamics. In 2020, DeMarco was elected a member
Laura_DeMarco
Munchausen number
while x not in cycle: cycle.append(x) x = pddif(x, b) return cycle Arithmetic dynamics Dudeney number Factorion Happy number Kaprekar's constant Kaprekar
Perfect digit-to-digit invariant
Perfect_digit-to-digit_invariant
Number that is its own Gödel number
reaches a fixed point. All numbers are in base b {\displaystyle b} . Arithmetic dynamics Dudeney number Factorion Happy number Kaprekar's constant Kaprekar
Meertens_number
Numbers whose sum of divisors is twice the number plus 1
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Quasiperfect_number
A non-constant polynomial with coefficients in a field is said to be eventually stable if the number of irreducible factors of the n {\displaystyle n}
Eventually_stable_polynomial
Mathematical formula
seen: seen.append(x) x = digit_product(x, b) return len(seen) - 1 Arithmetic dynamics Digit sum Digital root Sum-product number Weisstein, Eric W. "Multiplicative
Multiplicative_digital_root
Numbers whose aliquot sums form a cyclic sequence
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Sociable_number
Number that is the sum of the factorials of its digits
{\displaystyle m} . All numbers are represented in base b {\displaystyle b} . Arithmetic dynamics Dudeney number Happy number Kaprekar's constant Kaprekar number Meertens
Factorion
Number equal to the product of the sum and product of its digits
not in cycle: cycle.append(x) x = sum_product(x, b) return cycle Arithmetic dynamics Dudeney number Factorion Happy number Kaprekar's constant Kaprekar
Sum-product_number
Algebraic curve in mathematics
Tripling-oriented Doche–Icart–Kohel curve Jacobian curve Montgomery curve Arithmetic dynamics Elliptic algebra Elliptic surface Comparison of computer algebra
Elliptic_curve
Zero divisors in a module
may be computed in terms of division polynomials. Analytic torsion Arithmetic dynamics Flat module Annihilator (ring theory) Localization of a module Rank
Torsion_(algebra)
British mathematician
West Sussex – 30 July 2010) was a British mathematician working on arithmetic dynamics and recursive equations in number theory. Everest studied at Bedford
Graham_Everest
American mathematics professor
of the London Mathematical Society "for her deep contributions to arithmetic dynamics, to equidistribution, to bifurcation loci in families of rational
Holly_Krieger
Number that is the sum of its own digits, each raised to a given power
while x not in cycle: cycle.append(x) x = pdif(x, p, b) return cycle Arithmetic dynamics Dudeney number Factorion Happy number Kaprekar's constant Kaprekar
Perfect_digital_invariant
Type of positive integer pairs
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Betrothed_numbers
Parameter in the Mandelbrot set
= 23 {\displaystyle k=23} and period n = 2 {\displaystyle n=2} . Arithmetic dynamics Feigenbaum point Dendrite (mathematics) Diaz-Ruelas, A.; Baldovin
Misiurewicz_point
Numbers whose sum of divisors is twice the number minus 1
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Almost_perfect_number
Mixing (mathematics) Almost periodic function Symbolic dynamics Time scale calculus Arithmetic dynamics Sequential dynamical system Graph dynamical system
List of dynamical systems and differential equations topics
List_of_dynamical_systems_and_differential_equations_topics
Positional system with signed digits; the representation may not be unique
In mathematical notation for numbers, a signed-digit representation is a positional numeral system with a set of signed digits used to encode the integers
Signed-digit_representation
Doubly exponential integer sequence
MR 0384675. Jones, Rafe (2006). "The density of prime divisors in the arithmetic dynamics of quadratic polynomials". Journal of the London Mathematical Society
Sylvester's_sequence
Mathematics prize
Ana Caraiani, who was awarded the prize in 2025 "for contributions to arithmetic geometry and number theory: in particular, the Langlands program.". List
Ruth Lyttle Satter Prize in Mathematics
Ruth_Lyttle_Satter_Prize_in_Mathematics
Computer simulations to discover and understand chemical properties
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed
Molecular_dynamics
Number systems with a non-integer radix (base), such as base 2.5
Lectures Sidorov, Nikita (2003), "Arithmetic dynamics", in Bezuglyi, Sergey; Kolyada, Sergiy (eds.), Topics in dynamics and ergodic theory. Survey papers
Non-integer base of numeration
Non-integer_base_of_numeration
Number used for counting
numbers coming after smaller ones in the list 1, 2, 3, .... Two basic arithmetical operations are defined on natural numbers: addition and multiplication
Natural_number
American mathematician
Mathematical Society "for contributions to arithmetic potential theory, computational number theory, and arithmetic dynamics". "Robert Rumely", Mathematics Department
Robert_Rumely
Function whose domain is the positive integers
e ( x ) {\displaystyle \log _{e}(x)} . In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain
Arithmetic_function
Number divisible only by 1 and itself
Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be
Prime_number
Formulation of classical mechanics
Lagrangian mechanics is the Lagrangian, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has units of energy, but
Lagrangian_mechanics
American mathematician
Ferrero-Washington). More recently, Washington has published on arithmetic dynamics, sums of powers of primes, and Iwasawa invariants of non-cyclotomic
Lawrence_C._Washington
In algebraic geometry, a point with rational coordinates
over a finite field k has a k-rational point. Mathematics portal Arithmetic dynamics Birational geometry Functor represented by a scheme Hindry & Silverman
Rational_point
Chinese mathematician (born 1981)
number theory, arithmetic geometry, and automorphic forms. In particular, his work focuses on arithmetic intersection theory, algebraic dynamics, Diophantine
Xinyi_Yuan
Number raised to the third power
In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together. The cube of a number
Cube_(algebra)
Integer having a non-trivial divisor
order of the factors. This fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a
Composite_number
Numbers obtained by adding the two previous ones
} For a given n, this matrix can be computed in O(log n) arithmetic operations, using the exponentiation by squaring method. Taking the determinant
Fibonacci_sequence
example graph dynamical system. Symbolic dynamics Analytic combinatorics Combinatorics and physics Arithmetic dynamics Alsedà, Lluís; Libre, Jaume; Misiurewicz
Combinatorics and dynamical systems
Combinatorics_and_dynamical_systems
Count of the possible partitions of a set
doi:10.1017/S1757748900002334. Becker, H. W.; Riordan, John (1948). "The arithmetic of Bell and Stirling numbers". American Journal of Mathematics. 70 (2):
Bell_number
Physical theory describing classical fields
∇ × A . {\displaystyle \mathbf {B} =\nabla \times \mathbf {A} .} Fluid dynamics has fields of pressure, density, and flow rate that are connected by conservation
Classical_field_theory
Recursive integer sequence
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Catalan_number
Computer program to render and display many kinds of fractals
arithmetic (also known as fixed-point arithmetic), for faster rendering on computers without math coprocessors. Since then, floating-point arithmetic
Fractint
American mathematician
2004 and a Ph.D. in 2007 at Brown University; her dissertation, Arithmetic Dynamics of Rational Maps, was supervised by Joseph H. Silverman. After a
Michelle_Manes
Ten raised to an integer power
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Power_of_10
Number equal to the sum of its proper divisors
www-groups.dcs.st-and.ac.uk. Retrieved 9 May 2018. In Introduction to Arithmetic, Chapter 16, he says of perfect numbers, "There is a method of producing
Perfect_number
Product of two prime numbers
Sequences. OEIS Foundation. Nowicki, Andrzej (2013-07-01), Second numbers in arithmetic progressions, arXiv:1306.6424 Conway, J. H. (2008-06-18), Counting Groups:
Semiprime
Type of composite number with an even number of digits
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Vampire_number
Number that is the result of operation on its own digits
expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×, ÷), additive inverses, parentheses, exponentiation
Friedman_number
Integer where the average of its positive divisors is also an integer
theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic number because
Arithmetic_number
Formulation of classical mechanics using momenta
Classical Dynamics (Cambridge lecture notes), University of Cambridge, retrieved 27 October 2010 Hamilton, William Rowan, On a General Method in Dynamics, Trinity
Hamiltonian_mechanics
Composite number in number theory
number is a composite number n {\displaystyle n} which in modular arithmetic satisfies the congruence relation: b n ≡ b ( mod n ) {\displaystyle b^{n}\equiv
Carmichael_number
Figurate number
S2CID 53079729 Wikimedia Commons has media related to triangular numbers. "Arithmetic series", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Triangular
Triangular_number
Class of natural numbers with many divisors
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Superior highly composite number
Superior_highly_composite_number
Integer whose representation contains every digit in its number base
fictitious credit card numbers (while others use strings of zeroes). Several arithmetic properties of pandigital numbers have been studied, in particular regarding
Pandigital_number
Infinite integer series where the next number is the sum of the two preceding it
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Lucas_number
Integer filtered out using a sieve similar to that of Eratosthenes
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Lucky_number
Result of multiplying four instances of a number together
In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together: n4 = n × n × n × n. Fourth powers
Fourth_power
Framework for scoring a behavior's complexity
persons' performance of complex tasks. For example, a person cannot perform arithmetic until the numeral representations of numbers are learned, or a person
Model of hierarchical complexity
Model_of_hierarchical_complexity
French mathematician (1941–2020)
Graduate Center in 1999, Szpiro began working on new research in arithmetic dynamics. In 1987, Szpiro received the Prix Doistau–Blutel from the French
Lucien_Szpiro
ARITHMETIC DYNAMICS
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ARITHMETIC DYNAMICS
Female
English
English Shakespeare character name derived from Roman Latin Porcius, PORTIA means "pig." A moon of Uranus was given this name.
Girl/Female
Greek
Twilight.
Boy/Male
Hindu
Ice, Snow mountain
Boy/Male
Tamil
Goutheesh | கோஉஂதிஷ
Wisdom
Girl/Female
Muslim
Memory
Boy/Male
American, Anglo, British, English
From the Pointed Hill
Girl/Female
Tamil
Vasumathi | வாஸà¯à®®à®¾à®‚தீ
Golden Moon, Apsara of unequalled splendor
Male
Egyptian
, an Egyptian functionary.
Surname or Lastname
English, Scottish, and northern Irish
English, Scottish, and northern Irish : occupational name for a watchman or guard, from Norman French wardein (a derivative of warder ‘to guard’).English : habitational name from any of various places, for example in Bedfordshire, County Durham, Kent, Northumbria, and Northamptonshire, called Warden, from Old English weard ‘watch’ + dūn ‘hill’. Compare Wardlaw and Wardle 1.
Boy/Male
Aramaic Gaelic
Ploughman.
ARITHMETIC DYNAMICS
ARITHMETIC DYNAMICS
ARITHMETIC DYNAMICS
ARITHMETIC DYNAMICS
ARITHMETIC DYNAMICS
n.
That part of arithmetic which treats of adding numbers.
n.
Arithmetic.
adv.
The arithmetical character 0; a cipher. See Cipher.
n.
A book containing the principles of this science.
a.
Having equal differences; as, the terms of arithmetical progression are equidifferent.
v. t.
To subject to arithmetical division.
adv.
Conformably to the principles or methods of arithmetic.
v. i.
To use figures in a mathematical process; to do sums in arithmetic.
n.
A system of arithmetic, in which numbers are expressed in a scale of 60; logistic arithmetic.
n.
Arithmetical subtraction.
n.
The rule of three, in arithmetic, in which the three given terms, together with the one sought, are proportional.
v. i.
To perform the arithmetical operation of addition; as, he adds rapidly.
a.
Having an assignable arithmetical or numerical value or meaning; not imaginary.
n.
One skilled in arithmetic.
n.
The four "liberal arts," arithmetic, music, geometry, and astronomy; -- so called by the schoolmen. See Trivium.
a.
Sexagesimal, or made on the scale of 60; as, logistic, or sexagesimal, arithmetic.
n.
The science of numbers; the art of computation by figures.
a.
Of or pertaining to a unit or units; relating to unity; as, the unitary method in arithmetic.
a.
Of or pertaining to arithmetic; according to the rules or method of arithmetic.
v. t.
To subtract by arithmetical operation; to deduct.