In
mathematics, modular arithmetic is a system of
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...
for
integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
in his book ''
Disquisitiones Arithmeticae'', published in 1801.
A familiar use of modular arithmetic is in the
12-hour clock
The 12-hour clock is a time convention in which the 24 hours of the day are divided into two periods: a.m. (from Latin , translating to "before midday") and p.m. (from Latin , translating to "after midday"). For different opinions on represent ...
, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a
24-hour clock
The modern 24-hour clock, popularly referred to in the United States as military time, is the convention of timekeeping in which the day runs from midnight to midnight and is divided into 24 hours. This is indicated by the hours (and minutes) pass ...
is displayed "3:00" on a 12-hour clock.
Congruence
Given an
integer , called a modulus, two integers and are said to be congruent modulo , if is a
divisor of their difference (that is, if there is an integer such that ).
Congruence modulo is a
congruence relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done wi ...
, meaning that it is an
equivalence relation that is compatible with the operations of
addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
,
subtraction, and
multiplication. Congruence modulo is denoted:
:
The parentheses mean that applies to the entire equation, not just to the right-hand side (here, ). This notation is not to be confused with the notation (without parentheses), which refers to the
modulo operation
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is th ...
. Indeed, denotes the unique integer such that and
(that is, the remainder of
when divided by
).
The congruence relation may be rewritten as
:
explicitly showing its relationship with
Euclidean division. However, the here need not be the remainder of the division of by Instead, what the statement asserts is that and have the same remainder when divided by . That is,
:
:
where is the common remainder. Subtracting these two expressions, we recover the previous relation:
:
by setting
Examples
In modulus 12, one can assert that:
:
because , which is a multiple of 12. Another way to express this is to say that both 38 and 14 have the same remainder 2, when divided by 12.
The definition of congruence also applies to negative values. For example:
:
Properties
The congruence relation satisfies all the conditions of an
equivalence relation:
* Reflexivity:
* Symmetry: if for all , , and .
* Transitivity: If and , then
If and or if then:
* for any integer (compatibility with translation)
* for any integer (compatibility with scaling)
* for any integer
* (compatibility with addition)
* (compatibility with subtraction)
* (compatibility with multiplication)
* for any non-negative integer (compatibility with exponentiation)
* , for any
polynomial with integer coefficients (compatibility with polynomial evaluation)
If , then it is generally false that . However, the following is true:
* If where is
Euler's totient function, then —provided that is
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...
with .
For cancellation of common terms, we have the following rules:
* If , where is any integer, then
* If and is coprime with , then
* If and , then
The
modular multiplicative inverse is defined by the following rules:
* Existence: there exists an integer denoted such that if and only if is coprime with . This integer is called a ''modular multiplicative inverse'' of modulo .
* If and exists, then (compatibility with multiplicative inverse, and, if , uniqueness modulo )
* If and is coprime to , then the solution to this linear congruence is given by
The multiplicative inverse may be efficiently computed by solving
Bézout's equation for
—using the
Extended Euclidean algorithm.
In particular, if is a prime number, then is coprime with for every such that ; thus a multiplicative inverse exists for all that is not congruent to zero modulo .
Some of the more advanced properties of congruence relations are the following:
*
Fermat's little theorem: If is prime and does not divide , then .
*
Euler's theorem: If and are coprime, then , where is
Euler's totient function
* A simple consequence of Fermat's little theorem is that if is prime, then is the multiplicative inverse of . More generally, from Euler's theorem, if and are coprime, then .
* Another simple consequence is that if where is Euler's totient function, then provided is
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...
with .
*
Wilson's theorem: is prime if and only if .
*
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of the ...
: For any , and coprime , , there exists a unique such that and . In fact, where is the inverse of modulo and is the inverse of modulo .
*
Lagrange's theorem: The congruence , where is prime, and is a
polynomial with integer coefficients such that , has at most roots.
*
Primitive root modulo : A number is a primitive root modulo if, for every integer coprime to , there is an integer such that . A primitive root modulo exists if and only if is equal to or , where is an odd prime number and is a positive integer. If a primitive root modulo exists, then there are exactly such primitive roots, where is the Euler's totient function.
*
Quadratic residue: An integer is a quadratic residue modulo , if there exists an integer such that .
Euler's criterion asserts that, if is an odd prime, and is not a multiple of , then is a quadratic residue modulo if and only if
::
Congruence classes
Like any congruence relation, congruence modulo is an
equivalence relation, and the
equivalence class of the integer , denoted by , is the set . This set, consisting of all the integers congruent to modulo , is called the congruence class, residue class, or simply residue of the integer modulo . When the modulus is known from the context, that residue may also be denoted .
Residue systems
Each residue class modulo may be represented by any one of its members, although we usually represent each residue class by the smallest nonnegative integer which belongs to that class (since this is the proper remainder which results from division). Any two members of different residue classes modulo are incongruent modulo . Furthermore, every integer belongs to one and only one residue class modulo .
The set of integers is called the least residue system modulo . Any set of integers, no two of which are congruent modulo , is called a complete residue system modulo .
The least residue system is a complete residue system, and a complete residue system is simply a set containing precisely one
representative of each residue class modulo . For example. the least residue system modulo 4 is . Some other complete residue systems modulo 4 include:
*
*
*
*
*
*
Some sets which are ''not'' complete residue systems modulo 4 are:
*, since 6 is congruent to 22 modulo 4.
*, since a complete residue system modulo 4 must have exactly 4 incongruent residue classes.
Reduced residue systems
Given the
Euler's totient function , any set of integers that are
relatively prime to and mutually incongruent under modulus is called a reduced residue system modulo . The set from above, for example, is an instance of a reduced residue system modulo 4.
Integers modulo ''n''
The set of all
congruence classes of the integers for a modulus is called the ring of integers modulo , and is denoted
,
, or
. The notation
is, however, not recommended because it can be confused with the set of
-adic integers. The
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
is fundamental to various branches of mathematics (see below).
The set is defined for ''n'' > 0 as:
:
(When ,
is not an
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
; rather, it is
isomorphic to
, since .)
We define addition, subtraction, and multiplication on
by the following rules:
*
*
*
The verification that this is a proper definition uses the properties given before.
In this way,
becomes a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
. For example, in the ring
, we have
:
as in the arithmetic for the 24-hour clock.
We use the notation
because this is the
quotient ring of
by the
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
, a set containing all integers divisible by , where
is the
singleton set . Thus
is a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
when
is a
maximal ideal (i.e., when is prime).
This can also be constructed from the group
under the addition operation alone. The residue class is the group
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
of in the
quotient group , a
cyclic group.
Rather than excluding the special case , it is more useful to include
(which, as mentioned before, is isomorphic to the ring
of integers). In fact, this inclusion is useful when discussing the
characteristic of a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
.
The ring of integers modulo is a
finite field if and only if is
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
(this ensures that every nonzero element has a
multiplicative inverse). If
is a
prime power with ''k'' > 1, there exists a unique (up to isomorphism) finite field
with elements, but this is ''not''
, which fails to be a field because it has
zero-divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
s.
The
multiplicative subgroup of integers modulo ''n'' is denoted by
. This consists of
(where ''a''
is coprime to ''n''), which are precisely the classes possessing a multiplicative inverse. This forms a commutative
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
under multiplication, with order
.
Extension to real numbers
Applications
In theoretical mathematics, modular arithmetic is one of the foundations of
number theory, touching on almost every aspect of its study, and it is also used extensively in
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
,
ring theory,
knot theory
In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...
, and
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...
. In applied mathematics, it is used in
computer algebra
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressio ...
,
cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adve ...
,
computer science,
chemistry and the
visual
The visual system comprises the sensory organ (the eye) and parts of the central nervous system (the retina containing photoreceptor cells, the optic nerve, the optic tract and the visual cortex) which gives organisms the sense of sight ...
and
musical arts.
A very practical application is to calculate checksums within serial number identifiers. For example,
International Standard Book Number
The International Standard Book Number (ISBN) is a numeric commercial book identifier that is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.
An ISBN is assigned to each separate edition an ...
(ISBN) uses modulo 11 (for 10 digit ISBN) or modulo 10 (for 13 digit ISBN) arithmetic for error detection. Likewise,
International Bank Account Numbers (IBANs), for example, make use of modulo 97 arithmetic to spot user input errors in bank account numbers. In chemistry, the last digit of the
CAS registry number
A CAS Registry Number (also referred to as CAS RN or informally CAS Number) is a unique identification number assigned by the Chemical Abstracts Service (CAS), US to every chemical substance described in the open scientific literature. It inclu ...
(a unique identifying number for each chemical compound) is a
check digit
A check digit is a form of redundancy check used for error detection on identification numbers, such as bank account numbers, which are used in an application where they will at least sometimes be input manually. It is analogous to a binary parity ...
, which is calculated by taking the last digit of the first two parts of the CAS registry number times 1, the previous digit times 2, the previous digit times 3 etc., adding all these up and computing the sum modulo 10.
In cryptography, modular arithmetic directly underpins
public key systems such as
RSA and
Diffie–Hellman, and provides
finite fields which underlie
elliptic curves, and is used in a variety of
symmetric key algorithm
Symmetric-key algorithms are algorithms for cryptography that use the same cryptographic keys for both the encryption of plaintext and the decryption of ciphertext. The keys may be identical, or there may be a simple transformation to go between t ...
s including
Advanced Encryption Standard
The Advanced Encryption Standard (AES), also known by its original name Rijndael (), is a specification for the encryption of electronic data established by the U.S. National Institute of Standards and Technology (NIST) in 2001.
AES is a variant ...
(AES),
International Data Encryption Algorithm (IDEA), and
RC4
In cryptography, RC4 (Rivest Cipher 4, also known as ARC4 or ARCFOUR, meaning Alleged RC4, see below) is a stream cipher. While it is remarkable for its simplicity and speed in software, multiple vulnerabilities have been discovered in RC4, ren ...
. RSA and Diffie–Hellman use
modular exponentiation
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie-Hellman Key Exchange and RSA public/private keys.
Modular ...
.
In computer algebra, modular arithmetic is commonly used to limit the size of integer coefficients in intermediate calculations and data. It is used in
polynomial factorization, a problem for which all known efficient algorithms use modular arithmetic. It is used by the most efficient implementations of
polynomial greatest common divisor, exact
linear algebra and
Gröbner basis
In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Gröbne ...
algorithms over the integers and the rational numbers. As posted on
Fidonet in the 1980s and archived at
Rosetta Code
Rosetta Code is a wiki-based programming website with implementations of common algorithms and solutions to various programming problems in many different programming languages. It is named for the Rosetta Stone, which has the same text inscrib ...
, modular arithmetic was used to disprove
Euler's sum of powers conjecture on a
Sinclair QL microcomputer using just one-fourth of the integer precision used by a
CDC 6600
The CDC 6600 was the flagship of the 6000 series of mainframe computer systems manufactured by Control Data Corporation. Generally considered to be the first successful supercomputer, it outperformed the industry's prior recordholder, the IBM ...
supercomputer to disprove it two decades earlier via a
brute force search
In computer science, brute-force search or exhaustive search, also known as generate and test, is a very general problem-solving technique and algorithmic paradigm that consists of systematically enumerating all possible candidates for the solu ...
.
In computer science, modular arithmetic is often applied in
bitwise operation
In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic oper ...
s and other operations involving fixed-width, cyclic
data structures. The
modulo operation
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is th ...
, as implemented in many
programming languages and
calculator
An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics.
The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
s, is an application of modular arithmetic that is often used in this context. The logical operator
XOR sums 2 bits, modulo 2.
In music, arithmetic modulo 12 is used in the consideration of the system of
twelve-tone equal temperament, where
octave and
enharmonic equivalency occurs (that is, pitches in a 1:2 or 2:1 ratio are equivalent, and C-
sharp
Sharp or SHARP may refer to:
Acronyms
* SHARP (helmet ratings) (Safety Helmet Assessment and Rating Programme), a British motorcycle helmet safety rating scheme
* Self Help Addiction Recovery Program, a charitable organisation founded in 199 ...
is considered the same as D-
flat).
The method of
casting out nines
Casting out nines is any of three arithmetical procedures:
*Adding the decimal digits of a positive whole number, while optionally ignoring any 9s or digits which sum to a multiple of 9. The result of this procedure is a number which is smaller th ...
offers a quick check of decimal arithmetic computations performed by hand. It is based on modular arithmetic modulo 9, and specifically on the crucial property that 10 ≡ 1 (mod 9).
Arithmetic modulo 7 is used in algorithms that determine the day of the week for a given date. In particular,
Zeller's congruence and the
Doomsday algorithm
The Doomsday rule, Doomsday algorithm or Doomsday method is an algorithm of determination of the day of the week for a given date. It provides a perpetual calendar because the Gregorian calendar moves in cycles of 400 years. The algorithm for me ...
make heavy use of modulo-7 arithmetic.
More generally, modular arithmetic also has application in disciplines such as
law
Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vari ...
(e.g.,
apportionment
The legal term apportionment (french: apportionement; Mediaeval Latin: , derived from la, portio, share), also called delimitation, is in general the distribution or allotment of proper shares, though may have different meanings in different c ...
),
economics (e.g.,
game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applic ...
) and other areas of the
social sciences, where
proportional division and allocation of resources plays a central part of the analysis.
Computational complexity
Since modular arithmetic has such a wide range of applications, it is important to know how hard it is to solve a system of congruences. A linear system of congruences can be solved in
polynomial time with a form of
Gaussian elimination
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
, for details see
linear congruence theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
. Algorithms, such as
Montgomery reduction, also exist to allow simple arithmetic operations, such as multiplication and
exponentiation modulo , to be performed efficiently on large numbers.
Some operations, like finding a
discrete logarithm
In mathematics, for given real numbers ''a'' and ''b'', the logarithm log''b'' ''a'' is a number ''x'' such that . Analogously, in any group ''G'', powers ''b'k'' can be defined for all integers ''k'', and the discrete logarithm log'' ...
or a
quadratic congruence appear to be as hard as
integer factorization and thus are a starting point for
cryptographic algorithms
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adve ...
and
encryption. These problems might be
NP-intermediate.
Solving a system of non-linear modular arithmetic equations is
NP-complete.
Example implementations
Below are three reasonably fast C functions, two for performing modular multiplication and one for modular exponentiation on unsigned integers not larger than 63 bits, without overflow of the transient operations.
An algorithmic way to compute
:
[This code uses the C literal notation for unsigned long long hexadecimal numbers, which end with ]ULL
. See also section 6.4.4 of the language specificatio
n1570
uint64_t mul_mod(uint64_t a, uint64_t b, uint64_t m)
On computer architectures where an
extended precision
Extended precision refers to floating-point number formats that provide greater precision than the basic floating-point formats. Extended precision formats support a basic format by minimizing roundoff and overflow errors in intermediate values ...
format with at least 64 bits of mantissa is available (such as the
long double
In C and related programming languages, long double refers to a floating-point data type that is often more precise than double precision though the language standard only requires it to be at least as precise as double. As with C's other float ...
type of most x86 C compilers), the following routine is faster than a solution using a loop, by employing the trick that, by hardware,
floating-point
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be r ...
multiplication results in the most significant bits of the product kept, while integer multiplication results in the least significant bits kept:
uint64_t mul_mod(uint64_t a, uint64_t b, uint64_t m)
Below is a C function for performing modular exponentiation, that uses the function implemented above.
An algorithmic way to compute
:
uint64_t pow_mod(uint64_t a, uint64_t b, uint64_t m)
However, for all above routines to work, must not exceed 63 bits.
See also
*
Boolean ring In mathematics, a Boolean ring ''R'' is a ring for which ''x''2 = ''x'' for all ''x'' in ''R'', that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2.
Every Boolean ring gives rise to a Boolean al ...
*
Circular buffer
In computer science, a circular buffer, circular queue, cyclic buffer or ring buffer is a data structure that uses a single, fixed-size buffer as if it were connected end-to-end. This structure lends itself easily to buffering data streams. Ther ...
*
Division (mathematics)
*
Finite field
*
Legendre symbol
*
Modular exponentiation
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie-Hellman Key Exchange and RSA public/private keys.
Modular ...
*
Modulo (mathematics)
In mathematics, the term ''modulo'' ("with respect to a modulus of", the Latin ablative of '' modulus'' which itself means "a small measure") is often used to assert that two distinct mathematical objects can be regarded as equivalent—if their ...
*
Multiplicative group of integers modulo n
*
Pisano period
In number theory, the ''n''th Pisano period, written as '(''n''), is the period with which the sequence of Fibonacci numbers taken modulo ''n'' repeats. Pisano periods are named after Leonardo Pisano, better known as Fibonacci. The existence ...
(Fibonacci sequences modulo ''n'')
*
Primitive root modulo n
*
Quadratic reciprocity
*
Quadratic residue
*
Rational reconstruction (mathematics)
*
Reduced residue system
*
Serial number arithmetic (a special case of modular arithmetic)
*
Two-element Boolean algebra
* Topics relating to the group theory behind modular arithmetic:
**
Cyclic group
**
Multiplicative group of integers modulo n
* Other important theorems relating to modular arithmetic:
**
Carmichael's theorem
In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael,
states that, for any nondegenerate Lucas sequence of the first kind ''U'n''(''P'', ''Q'') with relatively prime parameters ''P'',&nbs ...
**
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of the ...
**
Euler's theorem
**
Fermat's little theorem (a special case of Euler's theorem)
**
Lagrange's theorem
**
Thue's lemma
Notes
References
* John L. Berggren
"modular arithmetic" Encyclopædia Britannica.
* . See in particular chapters 5 and 6 for a review of basic modular arithmetic.
* Maarten Bullynck
Modular Arithmetic before C.F. Gauss. Systematisations and discussions on remainder problems in 18th-century Germany
*
Thomas H. Cormen
Thomas H. Cormen is the co-author of ''Introduction to Algorithms'', along with Charles Leiserson, Ron Rivest, and Cliff Stein. In 2013, he published a new book titled '' Algorithms Unlocked''. He is a professor of computer science at Dartmou ...
,
Charles E. Leiserson
Charles Eric Leiserson is a computer scientist, specializing in the theory of parallel computing and distributed computing, and particularly practical applications thereof. As part of this effort, he developed the Cilk multithreaded language. H ...
,
Ronald L. Rivest
Ronald Linn Rivest (; born May 6, 1947) is a cryptographer and an Institute Professor at MIT. He is a member of MIT's Department of Electrical Engineering and Computer Science (EECS) and a member of MIT's Computer Science and Artificial Int ...
, and
Clifford Stein
Clifford Seth Stein (born December 14, 1965), a computer scientist, is a professor of industrial engineering and operations research at Columbia University in New York, NY, where he also holds an appointment in the Department of Computer Scie ...
. ''
Introduction to Algorithms'', Second Edition. MIT Press and McGraw-Hill, 2001. . Section 31.3: Modular arithmetic, pp. 862–868.
Anthony Gioia ''Number Theory, an Introduction'' Reprint (2001) Dover. .
*
*
*
External links
*
* In thi
article, one can learn more about applications of modular arithmetic in art.
* A
articleon modular arithmetic on the GIMPS wiki
Modular Arithmetic and patterns in addition and multiplication tables
{{Number theory
Finite rings
Group theory
Articles with example C code