Questions tagged [modular-arithmetic]
The modular-arithmetic tag has no usage guidance.
98
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Adding Points on an Elliptic Curve
How do I do this in Mathematica?
The curve given by the equation $y^2 = x^3 + 29x + 7$ defined over the integers modulo $10007$
Adding the first pair of points $P = (3737, 4549)$ and $Q = (8286, 6097)$...
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1
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0
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51
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How to solve quartic equation modulo a composite? [closed]
I have an univariate polynomial equation over a composite moduli.
Namely the composite is of for $q=(2p)^{2}-1$ where $p$ is odd and $2p-1$ and $2p+1$ are distinct primes.
The modular equation is
$$ax^...
- 145
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0
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59
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Weird behavior of repeated Mod of Hash
I was working on a wolfram challenge problem involving Hash, and I noticed some very odd behavior. The problem is to create a function ...
- 3,548
4
votes
1
answer
105
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Simplify expressions with modulo 2 (Boolean algebra)
Suppose I have a (Boolean algebra) expression
a = p1 x^2 + p2 x (x-1) + 3 p3 + 4 p4 x
where p1, ..., ...
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4
votes
1
answer
169
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How do I multiply two matrices mod2?
I have two matrices $A$ and $B$ which I multiply to give
$$\begin{bmatrix}
2 \\
2 \\
2
\end{bmatrix}$$
How do I multiply these matrices Modulo 2? I would like to end up with an all zero column vector.
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1
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3
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129
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2
votes
0
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27
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Effective way of iterating multiplication modulo p
I am trying to compute discrete logarithms modulo p (in a naive way). I have three approaches
...
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0
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28
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Polynomial GCD over a ring with composite modulus [duplicate]
I need the gcd of two polynomials,
...
3
votes
2
answers
173
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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 ...
4
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1
answer
184
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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
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76
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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 ...
- 3,907
0
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0
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72
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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 ...
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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
2
answers
469
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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*}$$
I tried it this ...
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2
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1
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48
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What is the meaning of this Modulus output?
Quick question, what is the meaning of the $\mathbb{c}_1$ subscript in the output of Modulus (shown as an image below)?
I understand there are two solutions to this linear congruence, 6 and 15 (6+9), ...
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1
answer
107
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Eliminate an expression with Mod operation
The target below should equal 0, since 2x + c = a + b + c = 6c, and Mod[6c,3] == 0
But ...
- 123
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1
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69
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Find a solution to a congruence within a specified range
My problem is to find all the integer solutions of the following conguence:
$$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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169
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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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3
votes
1
answer
309
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Plot solution to damped driven pendulum in $(-\pi,\pi)$
I'm solving the damped driven pendulum:
...
- 23.8k
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1
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84
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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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81
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3
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2
answers
466
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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
75
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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
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1
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78
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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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1
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48
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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
81
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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:
...
- 123
3
votes
1
answer
216
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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 ...
11
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3
answers
2k
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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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2
votes
1
answer
80
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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 ...
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0
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63
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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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0
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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
<...
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3
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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
...
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0
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67
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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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1
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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 }
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2
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1
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677
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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 ...
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votes
1
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157
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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
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1
answer
385
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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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3
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1
answer
257
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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 ...
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vote
1
answer
78
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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 ...
4
votes
3
answers
390
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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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1
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50
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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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2
votes
1
answer
255
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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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6
votes
1
answer
791
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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 ...
0
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174
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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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1
answer
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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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0
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1
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58
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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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7
votes
2
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2k
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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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4
votes
1
answer
358
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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
...
2
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1
answer
211
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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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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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