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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.
Modular arithmetic for a modulus defines any two elements and that differ by a multiple of to be equivalent, denoted by . Every integer is equivalent to one of the integers from 0 {\displaystyle 0} to n − 1 {\displaystyle n-1} , and the operations of modular arithmetic modify normal arithmetic by replacing the result of any ...
This result is the basis for rational reconstruction, which allows using modular arithmetic for computing rational numbers for which one knows bounds for numerators and denominators. [ 5 ] The proof is rather easy: by multiplying each congruence by the other y i and subtracting, one gets
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo of an odd prime number p: its value at a (nonzero) quadratic residue mod p is 1 and at a non-quadratic residue (non-residue) is −1.
In mathematics, modular arithmetic is a system of arithmetic for certain equivalence classes of integers, called congruence classes. Sometimes it is suggestively called 'clock arithmetic', where numbers 'wrap around' after they reach a certain value (the modulus). For example, when the modulus is 12, then any two numbers that leave the same ...
Hensel's original lemma concerns the relation between polynomial factorization over the integers and over the integers modulo a prime number p and its powers. It can be straightforwardly extended to the case where the integers are replaced by any commutative ring, and p is replaced by any maximal ideal (indeed, the maximal ideals of have the form , where p is a prime number).
The genus can be read off the field of meromorphic functions on the surface. Functions on a suitable topological space X into a field F can be added and multiplied pointwise, e.g., the product of two functions is defined by the product of their values within the domain: (f ⋅ g)(x) = f(x) ⋅ g(x). This makes these functions a F-commutative ...
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.
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