Modulo Properties (identities)
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- In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the modulus...
- In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable...
- number theory, a kth root of unity modulo n for positive integers k, n ≥ 2, is a root of unity in the ring of integers modulo n; that is, a solution x to the...
- Quadratic residue (redirect from Square root modulo n)theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that:...
Fibonacci sequence (section Periodicity modulo n)determined by the value of p modulo 5. If p is congruent to 1 or 4 modulo 5, then p divides Fp−1, and if p is congruent to 2 or 3 modulo 5, then, p divides Fp+1...
Hash function (section Properties)(the starting value could be 0, the identity value for XOR, or a random seed). The resulting value was reduced by modulo, folding or some other operation...- Ring (mathematics) (redirect from Arithmetic properties)familiar properties of addition and multiplication of integers. Some basic properties of a ring follow immediately from the axioms: The additive identity is...
Sine and cosine (section Identities)identities Madhava series Madhava's sine table Optical sine theorem Polar sine—a generalization to vertex angles Proofs of trigonometric identities Sinc...- GF(2) (section Properties)and multiplicative identities respectively denoted 0 and 1. Its addition is defined as the usual addition of integers but modulo 2 and corresponds to...
- modulo a prime number p, then this root can be lifted to a unique root modulo any higher power of p. More generally, if a polynomial factors modulo p...
- Finite field (redirect from Integers modulo a prime)addition and subtraction modulo q – 1. However, addition amounts to computing the discrete logarithm of am + an. The identity am + an = an(am−n + 1) allows...
- Determinant (redirect from Determinant identities)some of their properties. Namely, the determinant is the unique function defined on the n × n matrices that has the four following properties: The determinant...
- check digit method would be to take the sum of all digits (digital sum) modulo 10. This would catch any single-digit error, as such an error would always...
Complement (set theory) (section Properties)Let A and B be two sets in a universe U. The following identities capture important properties of absolute complements: De Morgan's laws: ( A ∪ B ) ∁...
Square (algebra) (section Related identities)particularly important in the finite fields Z/pZ formed by the numbers modulo an odd prime number p. A non-zero element of this field is called a quadratic...
P-adic number (section Topological properties)division by n, called its residue modulo n. The main property of modular arithmetic is that the residue modulo n of the result of a succession of operations...
Quotient group (section Example: Addition modulo 6)structure is "factored" out). For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying...- Discrete logarithm (section Properties)index: we can write x = indr a (mod m) (read "the index of a to the base r modulo m") for r x ≡ a (mod m) if r is a primitive root of m and gcd(a,m) = 1....
Isomorphism (section Integers modulo 6)1, or 2, where addition in the x-coordinate is modulo 2 and addition in the y-coordinate is modulo 3. These structures are isomorphic under addition...
Parity (mathematics) (section Properties)commutative and associative in modulo 2 arithmetic, and multiplication is distributive over addition. However, subtraction in modulo 2 is identical to addition...
- this field are the integers modulo p, and the field arithmetic is implemented in terms of the arithmetic of integers modulo p. A binary field is the field
- which the coefficients of the forms are either natural integers reduced modulo p, p prime, or elements of a Galois field. The linear transformations in
- n be a positive integer. Two integers a and b are said to be congruent modulo n, if their difference a − b is an integer multiple of n. If this is the