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 ...
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2 votes
0 answers
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
1 answer
58 views

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 ...
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  • 2,762
0 votes
0 answers
43 views

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 ...
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  • 101
0 votes
1 answer
52 views

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 ...
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1 vote
2 answers
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 ...
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  • 389
0 votes
0 answers
67 views

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 ...
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  • 443
2 votes
1 answer
41 views

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), ...
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0 votes
1 answer
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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0 votes
1 answer
41 views

Find a solution to a congruence within a specified range

My problem is to find all the integer solutions of the following conguence: $$P=0\ \text{mod}\ 4$$ where $$p=q^2∗r^2∗y^4∗z^2∗(2∗w^4+2∗r^2∗w^4∗y^4∗z^2+r^2∗w^8∗y^4∗z^2+2) + q^2+ 2∗r^6∗w^4∗y^m∗z^6∗(w+w^2+...
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  • 121
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0 answers
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Command to find the pre-image of a set that is mapped to by the function $f(x)=3^x \mod{17}$

$$f(x)=3^x \mod{17}$$ How to find the pre-image of the set if this function maps from real numbers to real numbers. What command would I use, to see the pre-image if the function would map from ...
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  • 241
2 votes
1 answer
125 views

Plot solution to damped driven pendulum in $(-\pi,\pi)$

I'm solving the damped driven pendulum: ...
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  • 23.3k
0 votes
1 answer
73 views

Modulus option quietly nerfed for Solve?

Recently asked my students to put together something like this: Reduce[x^2 ==b, b, Modulus -> 11] For me, on 12.1, this gives: ...
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  • 2,576
0 votes
0 answers
55 views

PowerMod solve for b (exponent)

Using PowerMod[a, b, m] = x I get ...
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2 votes
2 answers
344 views

How to ask Mathematica to solve a simple modular equation

What's the most straightforward way to ask Mathematica to find all solutions of an equation like $$3x + 2y + 4z = 0 \pmod {11}$$ (for instance), where either $x$, $y$, $z$ can be considered to be ...
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3 votes
1 answer
60 views

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, ...
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1 vote
1 answer
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 ...
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0 votes
1 answer
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 ...
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2 votes
1 answer
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: ...
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2 votes
1 answer
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 ...
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11 votes
3 answers
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) $\bmod n$...
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  • 333
2 votes
1 answer
68 views

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 ...
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  • 753
1 vote
0 answers
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 ...
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  • 1,095
2 votes
0 answers
48 views

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 <...
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  • 2,020
4 votes
3 answers
115 views

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 ...
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  • 6,827
0 votes
0 answers
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 ...
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  • 14.8k
5 votes
1 answer
680 views

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 }
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  • 153
2 votes
1 answer
470 views

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 ...
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5 votes
1 answer
116 views

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 ...
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1 vote
1 answer
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 ...
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  • 2,842
3 votes
1 answer
229 views

List of invertible congruence classes

I am attempting to create a list of the invertible congruence classes $\bmod 120$. The code I have is ...
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  • 171
1 vote
1 answer
71 views

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 ...
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4 votes
3 answers
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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  • 8,377
0 votes
1 answer
47 views

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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2 votes
1 answer
212 views

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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6 votes
1 answer
625 views

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 ...
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0 votes
0 answers
136 views

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

I have the following $2$-dimensional implicitly defined map ...
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  • 211
1 vote
1 answer
128 views

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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0 votes
1 answer
55 views

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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7 votes
2 answers
2k views

How can I solve a certain congruence equation?

I want to solve the congruence $3x^2 + 6x + 1 \equiv 0 \pmod {19}$. I know that one way to solve this is to first solve the congruence: $t^2 \equiv 5 \pmod {19}$, because $b^2 - 4ac \equiv 24 \equiv ...
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  • 71
4 votes
1 answer
305 views

Matrix operations "Modulus->non-prime" problem

I am using Mathematica 10.4.1.0. When I specify the "Modulus" option to be a non-prime, the functions RowReduce and MatrixRank return the error ...
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2 votes
1 answer
185 views

How to remove symmetries from this list?

First of all, I work with modular arithmetics and the elements of the arithmetics are lists themselves. I recognize two kinds of symmetries - shift and mirroring. We could define shift of a list like ...
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  • 325
9 votes
1 answer
194 views

Modular equation problem

I have problem solving this modular equation $67^n \equiv 67 \pmod {317026939759222841944}$ with $n>1$. I have tried my Laptop and Wolfram Alpha engine, but I don't get any solution, I'm very ...
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1 vote
0 answers
71 views

Last digits via PowerMod [closed]

Do you have an idea why this produces different results? PowerMod[2003, 2002^2001, 1000] 241 ...
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  • 5,729
6 votes
2 answers
204 views

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 ...
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  • 163
0 votes
0 answers
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}} ...
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  • 14.6k
2 votes
0 answers
110 views

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 ...
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  • 121
0 votes
1 answer
61 views

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: <...
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  • 3
3 votes
1 answer
225 views

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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3 votes
2 answers
405 views

Evaluate modular fractions

If I have a list of numbers l = {256/11, 258/11, 263/11, 263/11, -22, -22, 251/11, 0, 261/11, -22, 265/11, 259/11, 0, 259/11} which I want to evaluate mod 26, how ...
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