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In mathematics, modular arithmetic is a system of arithmetic 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 in his book Disquisitiones Arithmeticae , published in 1801.
In mathematics, the modular group is the projective special linear group (,) of matrices with integer coefficients and determinant, such that the matrices and are identified. The modular group acts on the upper-half of the complex plane by linear fractional transformations .
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 a and n , a modulo n (often abbreviated as a mod n ) is the remainder of the Euclidean division of a by n , where a is the dividend and n is the divisor .
The above definition states that a composite integer n is Carmichael precisely when the nth-power-raising function p n from the ring Z n of integers modulo n to itself is the identity function. The identity is the only Z n-algebra endomorphism on Z n so we can restate the definition as asking that p n be an algebra endomorphism of Z n.
Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. [3] Given the integers a, b and n, the expression "a ≡ b (mod n)", pronounced "a is congruent to b modulo n", means that a − b is an integer multiple of n, or equivalently, a and b both share the same remainder when divided by n.
The original proof of Thue's lemma is not efficient, in the sense that it does not provide any fast method for computing the solution. The extended Euclidean algorithm, allows us to provide a proof that leads to an efficient algorithm that has the same computational complexity of the Euclidean algorithm.
The corresponding addition and multiplication of equivalence classes is known as modular arithmetic. From the point of view of abstract algebra, congruence modulo n {\displaystyle n} is a congruence relation on the ring of integers, and arithmetic modulo n {\displaystyle n} occurs on the corresponding quotient ring .
The ray class group modulo m is the quotient C m = I m / i(K m,1). [ 14 ] [ 15 ] A coset of i( K m ,1 ) is called a ray class modulo m . Erich Hecke 's original definition of Hecke characters may be interpreted in terms of characters of the ray class group with respect to some modulus m .
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