Mod

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

General characteristics

Domain and analyticity

m mod n is a nonanalytical function; it is a piecewise continuous function which is defined over (ℂ^2).

Symmetries and periodicities

Parity

m mod n is an odd function.

Mirror symmetry

Periodicity

m mod n is a periodic function with respect to m with period n.

Sets of discontinuity

The function  m mod n  is a piecewise continuous function with jumps on the curves Re((m/n))=k∨Im((m/n))=l/;k,l∈ℤ. The functional property m mod n==n ((m/n) mod 1)==n ((m/n)-⌊(m/n)⌋) makes the behaviour of the m mod n  similar to the behaviour of ⌊(m/n)⌋.

Series representations

Exponential Fourier series

Other series representations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Argument involving related functions

Addition formulas

Multiple arguments

Related transformations

Identities

Functional identities

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

With respect to m

With respect to n

Fractional integro-differentiation

With respect to m

With respect to n

Integration

Indefinite integration

Involving only one direct function with respect to m

Involving one direct function and elementary functions with respect to m

Involving only one direct function with respect to n

Involving one direct function and elementary functions with respect to n

Definite integration

For the direct function with respect to m

In the following formulas a∈ℝ .

For the direct function with respect to n

In the following formulas a∈ℝ .

Integral transforms

Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Mellin transforms

Representations through equivalent functions

With related functions

With Floor

With Round

With Ceiling

With IntegerPart

With FractionalPart

With Quotient

With elementary functions

Zeros

Theorems

Linear congruential random number generator

A sequence of pseudorandom numbers (Subscript[r, k]) is generated by (Subscript[r, k+1])==(a (Subscript[r, k ])+c)mod m, with  a,c,m∈ℕ, m>0, 0≤a<m, 0≤c<m,0≤(Subscript[r, 0])<m.

Chinese remainder theorem

Let (Subscript[m, 1]), (Subscript[m, 2]), …, (Subscript[m, n])be pairwise relatively prime integers (gcd((Subscript[m, i]),(Subscript[m, k]))==1/;i≠k). Then for given integers (Subscript[z, 1]), (Subscript[z, 2]), …, (Subscript[z, n]) there exists a unique (mod (Subscript[m, k])) integer z such that z mod (Subscript[m, k])==(Subscript[z, k]).

Legendre theorem

If a, b, c∈(SuperPlus[ℕ])∧ gcd(a,b)==gcd(a,c)==gcd(b,c)==0∧(Sqrt[a]),(Sqrt[b]),(Sqrt[c])∉ℕ , then the equation a (x^2)+b (x^2)+c (z^2)==0 has nontrivial solutions for x,  y,  z if and only if the equations (x^2)- b c mod a==0∧(y^2)- a c mod b==0∧(z^2)- a b mod c==0 are solvable.

Gauss'  Easter formula

Easter Sunday is the i+j+1 th day after the 21st of March, where

i==28(Subscript[δ, h,29])+27(Subscript[δ, h,28])θ(a-11)+11(1-(Subscript[δ, h,29]))(1-(Subscript[δ, h,28])θ(a-11)),j==(2b+4c+6i+g)mod 7,h==(f+19a)mod 30,  a==year mod 19,b==year mod 4,c==year mod 7,  d==⌊(((8[year/100]+13))/25)⌋-2,e==⌊(year/100)⌋-⌊(year/400)⌋-2,f==(15+e-d)mod 30, g==(6+e)mod 7.

History

– C. F. Gauss (1801) introduced the symbol mod

Applications include pseudo-random number generation.

Copyright