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Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. 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 ...
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. [ 1 ] For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0 ...
The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Zn; it has φ (n) elements, φ being Euler's totient function, and is denoted as U (n) or ...
The formula was defined by Jeff Tupper and appears as an example in Tupper's 2001 SIGGRAPH paper on reliable two-dimensional computer graphing algorithms. [1] This paper discusses methods related to the GrafEq formula-graphing program developed by Tupper. [2] Although the formula is called "self-referential", Tupper did not name it as such. [3]
A prime number p = 2 q + 1 is called a safe prime if q is prime. Thus, p = 2 q + 1 is a safe prime if and only if q is a Sophie Germain prime, so finding safe primes and finding Sophie Germain primes are equivalent in computational difficulty. The notion of a safe prime can be strengthened to a strong prime, for which both p − 1 and p + 1 ...
If p ≡ 3 mod 4, that is p = 3, 7, 11, 19, ..., one may choose −1 ≡ p − 1 as a quadratic non-residue, which allows us to have a very simple irreducible polynomial X 2 + 1. Having chosen a quadratic non-residue r , let α be a symbolic square root of r , that is, a symbol that has the property α 2 = r , in the same way that the complex ...
Modular exponentiation is the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c = be mod m. From the definition of division, it follows that 0 ≤ c < m. For example, given b = 5, e = 3 and m = 13, dividing 53 = 125 by 13 leaves a remainder of c = 8.
In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a rational multiple thereof.