# Questions tagged [modular-arithmetic]

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### System of equations

I am trying to find ALL or a list of possible values for x satisfying: x MOD 9 ==2 AND (x+631) MOD 9 = 3 Using Mathematica. can anyone help? 2 is not the only answer 497 is another one since 497 mod ...
72 views

### Data visualization in Mathematica (1000 x 1000 square)

I'm currently interested in looking at all the 6-digit numbers in base 10 (i.e. 000000 to 99999), and checking if they satisfy at least one of 5 equations: either 0=0+0+0+0+0, 0+0=0+0+0+0,..., 0+0+0+0+...
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1 vote
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### Simplify of Mod in compound logical expression

It's great that Simplify[Mod[x,2π]==0,Assumptions->{0<=x<2π}] yields x==0. However even a slightly more complicated ...
• 2,762
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### Fourier expansion of Dedekind eta function with rational arguments

I want to be able to compute the Fourier series expansion in q=Exp[2πiz] for DedekindEta[(az+b)/(cz+d)] with integer a,b,c,d, but not requiring ad-bc=1. Just replacing z in terms of q and doing ...
• 101
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### ListPlot vs. ListLinePlot How Do You Remove Discontinuous Artefacts from Introduced by Mod?

I have some data that inherently describes a phase and would like to plot it in between [0, 2*pi). When I use ListPlot with the ...
• 419
1 vote
179 views

### Solve System of congruences

How can I solve a system of linear congruences as such? \begin{align*} 3x+2y+28z &= 9 \pmod {29} \\ 5x+27y+z &= 9 \pmod {29} \\ 2x+y+z &= 6 \pmod {29} \end{align*} I tried it this ...
• 389
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### Reduce does not work with Mod operation

We have a constant issue about working with some functions like Mod and Log. here when I add a ...
• 443
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### What is the meaning of this Modulus output?

Quick question, what is the meaning of the $\mathbb{c}_1$ subscript in the output of Modulus (shown as an image below)? I understand there are two solutions to this linear congruence, 6 and 15 (6+9), ...
101 views

### Eliminate an expression with Mod operation

The target below should equal 0, since 2x + c = a + b + c = 6c, and Mod[6c,3] == 0 But ...
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### Can Mathematica automatically optimize an expression with PowerMod?

PowerMod[x,p,m] is significantly faster than Mod[x^p, m] for a large p I have an expression, ...
1 vote
70 views

### A modular version of LinearRecurrence?

I'm playing around with implementing the Lucas probable prime test (mainly so I can understand it better), and would love a version of the LinearRecurrence function that used addition modulo n (with n ...
45 views

### Find relationship between values within finate range of integer domain

Update: There are some misakes in the coefficient of left and right. But still get similar result. I understand the mathematica ...
58 views

### How to simplify quotient and mod in integer domain

I want to simplify (a*32 + a%32)/32, and a is integer. So I write: ...
132 views

### Non-valid modulus when using LinearSolve

I have a system of linear equations I want to solve mod 4, and I happen to know the solution, but I get an error when trying to solve it using LinearSolve. I define the matrix ...
1k views

### How can I program the last digits of Fibonacci numbers?

Is there a way to compute the last digits of an arbitrarily large Fibonacci number? For the $10^n$th Fibonacci number, we can just find the $2^n$-th Fibonacci number (if that isn't too large) $\bmod n$...
• 333
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### How to get the modulus of two functions [closed]

I might be overlooking something, because what I want to do is quite elementary. I want the following term Mod[2(1 + z), (1 + z)] to evaluate to ...
• 753
1 vote
58 views

### How to apply PolynomialMod to a SparseArray?

I have a list bdrs of SparseArrays, whose entries are polynomials in the variable t with ...
• 1,095
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### Why gives Modulo of complex numbers different answers for rationals and reals

When running Table[1.*Mod[(1 + I)^(1 + k*I), 1 + k*I], {k, 49, 50, 0.5}] Table[1.*Mod[(1 + I)^(1 + k*I), 1 + k*I], {k, 49, 50, 1/2}] I get for the first table <...
• 2,020
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### how can I calculate $a_{n+3}=a_n+a_{n+1}+a_{n+2}$?

recently I need to calculate this: $a(n+1)=((a(n-2)+a(n-1)+a(n)) \bmod 10000)$ and get $a(20000000)$ (for example). I know RecurrenceTable,but ...
• 6,827
58 views

### Finding a solution to a modular problem efficiently

I want to find a naïve hash function of the form Mod[a x + b, 2^p] which would produce a value less or equal to v for all ...
• 14.8k
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### modulo with n instead of 0 without using if-statement

Is there a way to to express the following without using an if statement, but with mathematical operations instead? { 5 if x % 5 = 0, x % 5 otherwise }
• 153
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### How to Plot modulo function Mod on a function of two variables? [closed]

I know that Plot[Mod[x, 2], {x, -6, 6}] is a Mathematica command to plot the remainder of x over 2 when I range x over reals -6 to 6. How do I plot the same ...
• 123
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### FindEquationalProof to prove divisor theorem

I'm trying to implement proofs of concepts for Equational Proofs on some basic number theory theorems. One such example is: "let a and b be positive integers and let d = gcd (a, b). If t divides ...
1 vote
297 views

### Polynomial PowerMod

Is there equivalent of PowerMod for polynomials in Mathematica? We have Mod[a^e,m]==PowerMod[a,e,m], $a$, $e$ and $m$ all ...
• 2,842
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### List of invertible congruence classes

I am attempting to create a list of the invertible congruence classes $\bmod 120$. The code I have is ...
• 171
1 vote
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### Is this decryption function idiomatic?

This is my first non-trivial Mathematica notebook, which I wrote for a cryptology homework. It encrypts/decrypts strings using an affine cipher. Is the code readable/idiomatic? How could it be ...
378 views

### How to reverse in modular arithmetic

Given an RGB color {127, 255, 212} where scale is 0-255, VBA's RGB function find a long integer x such that ...
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### Looping on ranges (Modular iterations)

Is it possible to make the following code modular, so that it works up till depth n, and returns an array of all possible combinations for used{}? ...
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### How to solve a system with GCD?

For example, Solve[{x + y == 150, GCD[x, y] == 30}, {x, y}, Integers] Solve::nsmet: This system cannot be solved with the methods available to Solve. The same ...
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### RowReduce modulo 3

When I wanted to compute a row echelon form of a matrix over the finite field ${\Bbb F}_9$ with 9 elements (which is the three element field ${\Bbb F}_3$ adjoined a square root of $-1$ which I denote ...
136 views

### Variables defined modulo $\pi$ and avoiding singularities; phase plotting for a map

I have the following $2$-dimensional implicitly defined map ...
• 211
1 vote
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### Solving a modular equation with a large modulus [duplicate]

I am having trouble solving this equation: $17^n\equiv 1 \mod 640826899755722841329632946651705039010285107743541637238400825304943424424715705661705191669759$ for the smallest integer $n > 0$ I ...
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### Get missing cyclic numbers via modulus?

Consider a list of numbers: list = {1,2,3,4,5}; Taking a random set of three of these numbers: ...
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• 91
1 vote
71 views

### Last digits via PowerMod [closed]

Do you have an idea why this produces different results? PowerMod[2003, 2002^2001, 1000] 241 ...
• 5,729
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### Memory problem when solving a system of modular equations [duplicate]

I need to solve the following system of modular equations, but the computation can't finish because I run out of memory (I have 12 GB of RAM). Is there any workaround to this problem? I am using ...
• 163
54 views

### Function to add two matrices mod 2 [duplicate]

I have: lightstep[m_, gg_] := Which[gg == {1, 1}, m = Mod[m + m, 2]] Then I entered: A = {{1, 1, 0}, {1, 0, 0}, {0, 0, 0}} ...
• 14.6k
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### Finite fields package doesn't simplify

I'm trying to do arithmetic using the FiniteFields package. Supposing a is a generator of a Galois field (say GF(2^5)) I want to be able to simplify things like a^32 and (a^8)^-1. I don't care about ...
• 121
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### Putting some calculations together to form an algorithm

Considering these below where n is an odd number, k is also an odd number less than n: <...
• 3
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### Coppersmith method of small integer solutions to multivariate polynomials

In http://link.springer.com/chapter/10.1007%2F3-540-68339-9_14#page-7 is presented a method for finding small solutions to multivariate polynomials. (Although here he mentions that it "is not rigorous"...
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