Questions on the number-theoretic functionality of Mathematica.

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3
votes
5answers
1k views

A question regarding 1 divided 243

Here is a problem due to Feynman. If you take 1 divided by 243 you get 0.004115226337 .... It goes a little cockeyed after 559 when you're carrying out the decimal expansion, but it soon straightens ...
6
votes
5answers
771 views

Write any positive integer as a sum of squares

With n = 17 I would like to get {4, 1} and with n = 999 {31, 6, 1, 1} so that, for example, ...
2
votes
1answer
105 views

Symbolic multiplicative partitions

Let $p_n\#\equiv\prod_{k=1}^{n}p_k$ (primorial): p[n_] := Times @@ Prime[Range[n]] then the multiplicative partitions of $p_{1,2,3,4}\#$ are $$ \{\{2\}\},$$$$ ...
36
votes
5answers
2k views

Factorisation diagrams

Here is a way to visualize the factorisation of natural numbers. How do we get this or a similar kind of output using Mathematica? See the list of images generated for number from 1 to 36:
0
votes
2answers
117 views

Testing the Erdos square free conjecture

Can someone please answer these two parts of my homework assignment? Adding an explanation would be appreciated! a. Test the Erdos square free conjecture for $n <= 30000$. You should use ...
0
votes
0answers
253 views

Elliptic curve cryptography in Mathematica

I can find no resources for doing elliptic curve cryptography. I have used the finite field package, but I find it cumbersome and it does not seem to have any builtin methods for ECC. How can I get ...
0
votes
1answer
64 views

Square root of a value defined in a finite field?

I am trying to find the right way to compute the square root of a number defined in a finite field. For example, ...
3
votes
1answer
105 views
4
votes
0answers
91 views

Simplifying expressions involving Divisible

FullSimplify[ Divisible[p^2 - 1, 24] , Element[p, Primes] && p > 3] Should evaluate to True, but I get ...
2
votes
2answers
245 views

Checking if a number is a perfect power

I wanted to know how would I use Mathematica in order to check if the number is a perfect power I saw the algorithm but couldn't grasp it enough to implement it, so can anybody help?
4
votes
2answers
162 views

FactorInteger over UFDs

How can I factor 'integers' over other quadratic number fields (not just gaussian integers). For instance, how could I factor $7 = (3 + ω)(2 − ω)$ over Eisenstein integers ($ω = \frac{-1+ I ...
14
votes
4answers
2k views

Semi prime numbers

The high school textbook I am using has the example of semi-prime numbers. They wanted students to find (by "perspiration") all the semi-prime numbers less than $50$ (for a question on set theory). ...
1
vote
3answers
217 views

Generating a list of cubefree numbers

I am trying to generate a list of cubefree numbers (i.e. numbers when prime factorized contain no tripled factors) within a given range. Of DivisorSigma, ...
12
votes
6answers
2k views

How to find lattice points on a line segment?

How do I find points on the line segment joining {-4, 11} and {16, -1} whose coordinates are positive integers?
1
vote
1answer
258 views

Faster GCD Implementation

Is there any chance to write a faster GCD than the built-in one in Mathematica? @Mr.Wizard has written one in this question (although it's not for this purpose) which is 6 times slower on a 100k ...
8
votes
4answers
336 views

Generate PrimePower counting function

Is there a way to generate a counting function for prime powers - i.e. to create a similar function to PrimePi, but including prime powers. The following will, of ...
3
votes
1answer
250 views

Negative Continued Fraction of a Rational

The $n^{\text{th}}$ negative continued fraction convergent $x_n$ of a positive real $x$ is computed by the nested function \begin{align} x_n = k_1 - \frac{1}{k_2 - \frac{1}{k_3 - \dots - ...
4
votes
1answer
394 views

Von Mangoldt function

Can anybody evaluate the following sum for me $$ \sum\limits_{n=2}^\infty(-1)^n\left(\frac{\psi(n)}{n}-\frac{\Lambda(n)}{2n}\right) $$ where $\psi(n)$ is the Chebyshev function and $\Lambda(n)$ is ...
0
votes
0answers
121 views

How can I calculate all irreducible polynomials of 31 degree in $\mathbb Z_2[x]$?

How can I calculate all binary irreducible polynomials of degree 31? or how i calculate all irreducible $f$ in $\mathbb Z_2[x]$? (The irreducible polynomial in $\mathbb Z_2[x]$ and $\mathbb R$ are ...
1
vote
3answers
166 views

Number theory: Problem involving rational numbers

Use RandomRat to test whether ((-1)^(1/Denominator[q]))^Numerator[q] is identical with ...
5
votes
4answers
582 views

Perfect numbers

The question given to me: a. Find the perfect numbers between $1$ and $10^6$ b: Find the abundant numbers between $1$ and $1000$ For a, I wrote ...
3
votes
3answers
207 views

What is the formula for this numerical series?

I'm developing a questions game. My goal is that the score for each correct answer will increase as the user answers more questions. Initially there are 15 points for each correct answer. Every 4 ...
12
votes
5answers
692 views

Double series over primes

I'm very curious if the following double series over primes has a closed form: $$\sum_{k \in \mathcal{P}}\sum_{n \in \mathcal{P}}\frac{1}{k\;n(k+n)^2}$$ where $\mathcal{P}$ denotes the set of all ...
4
votes
1answer
459 views

Implementation of the Polynomial Chinese Remainder Theorem

I would like an implementation of the Chinese Remainder Theorem for polynomials in $\mathbb{Z}[x]$, that is, a function ...
-1
votes
3answers
289 views

Find integer values of p such that $(2^p - (2^2)(3^2))/ (3^3)$ is an integer

Find integer values of p such that $(2^p - (2^2)(3^2))/ (3^3)$ is an integer.
-5
votes
1answer
372 views

random number visualization without generator (Spectral Test)

Does anyone have codes for using only numbers to make this kind of random-number visualization and not requiring a number generator? ...
2
votes
1answer
249 views

Expressing large numbers in index form

I have a quick question. Is there anyway of expressing a large number as a power of another number in Mathematica? By this, I mean for example, $1237940039285380274899124224 = 512^{10}$. Is there a ...
14
votes
3answers
675 views

How to know if a number is the square of a rational?

I'm pretty new with Mathematica and I was looking for a way to know whether a number is a square of a rational. I thought of Head[Sqrt[myNumber]] == Rational ...
5
votes
1answer
166 views

How can I program the RiemannR function using the LogIntegral command?

I would like to program the RiemannR function using the LogIntegral command because I would like to later experiment with a ...
1
vote
1answer
209 views

Hermite Normal Form in “columns” convention

After describing the Hermite Normal Form (HNF), MathWorld explains: The Hermite normal form for integer matrices is implemented in Mathematica as ...
3
votes
1answer
661 views

Function to Determine Lucky Numbers

Given a list of the form {1, 3, 5, 7, ...}, the lucky numbers are obtained by looking at the first list element after 1 (so 3 in this case), and deleting all list ...
2
votes
3answers
545 views

Generating pairs of additive and multiplicative factors for integers

Given an integer $n$, I want to get two lists: a) the set of pairs of the divsors $a,b$ into exactly two factors $n=a\cdot b$, b) the set of pairs $a,b$ of two summands $n=a+b$. The code I ...
3
votes
1answer
188 views

Finding the largest integer that cannot be partitioned in a certain way

I want to use Mathematica to solve the problem: Find the maximum $k$ such that $6x+9y+20z=k$ does not have a non-negative solution. I tried FrobeniusSolve. ...
11
votes
2answers
537 views

FiniteFields package is very slow. Any fast substitute for Mathematica?

I want to compute the inverse of matrix, say with dimensions $100 \times 100$, defined over a large finite field extension such as $GF(2^{120})$. I am using the package FiniteFields, but Mathematica's ...
4
votes
1answer
419 views

How does Mathematica calculate the nth prime?

When I enter Prime[2000000000000], the two-trillionth prime, Mathematica gives 61427839512211 for the answer after several ...
2
votes
3answers
214 views

Why do these two different zetas produce the same value?

Zeta[-13] == Zeta[-1] == -1/12 Why do these two different zetas produce the same value?
3
votes
1answer
227 views

Generating a list of all factorizations

What is the best way to generate a list of all factorizations of some number $n$? I'm quite new to Mathematica so this might be obvious. I have been trying some basic stuff with ...
1
vote
1answer
141 views

Another MoebiusMu question

When I evaluate the Mertens function to infinity: NSum[MoebiusMu[k], {k, 1, \[Infinity]}] I get -1, but I expected to get -2. I wanted to modify the ...
6
votes
3answers
659 views

Implementing the Farey sequence efficiently

There is of course the silly implementation: FareySequence[n_] := Union[Flatten[Table[j/i, {i, 1, n}, {j, 0, i}]]] However, there are numerous properties and ...
2
votes
2answers
357 views

Plotting Chebyshev's theta function $\vartheta(x)$

The function I would like to plot is defined as $\sum\limits_{p\leq x}\log p.$ The following gives me I think a plot of the points of interest, but the function is defined for all $x > 0$ and so ...
5
votes
1answer
336 views

Which DirichletCharacter is KroneckerSymbol?

If $d$ is a fundamental discriminant, KroneckerSymbol[d,n] is a Dirichlet character modulo $|d|$. Which one is it? If $d>0$ is a prime $\equiv 1\bmod 4$, then ...
8
votes
1answer
208 views

Testing for primality in quadratic rings?

Testing for primality in $\mathbb{Z}[\sqrt{-1}]$ in Mathematica is easy: PrimeQ[n, GaussianIntegers -> True] But how can I test for primality in, say, ...
6
votes
1answer
150 views

Implementing Remainder Tree

I want to implement Remainder Tree based on this. With the answers on SE I've come up with: ...
14
votes
2answers
457 views

Why does iterating Prime in reverse order require much more time?

Say I would like to display the $10$ greatest primes that are less than $10^5$. I could do the following: ...
3
votes
0answers
278 views

Parallel PowerMod

Is there anyway to parallelize the PowerMod function? Here is my Left-To-Right modular exponentation: ...
6
votes
1answer
594 views

Evaluate continued fraction

Mathematica has the ContinuedFraction[] function to give the continued fraction expansion of a rational (or approximation of a real) number. I'm interested in the ...
6
votes
0answers
112 views

Doing computations in a modulo ring

I need to perform some computations in a modulo ring, like Mod[Subfactorial[n], m] Mod[Binomial[n, k], m] However, this is obviously much too slow for large ...
14
votes
5answers
2k views

Fastest square number test

What is the fastest possible square number test in Mathematica 7, both for machine size and big integers? I presume in version 8 the fastest will be a dedicated C LibraryLink function.
23
votes
6answers
2k views

efficient way to count the number of zeros at the (right) end of a very large number

If I want to count the number of zeros at the (right) end of a large number, like $12345!$, I can use something like: Length[Last[Split[IntegerDigits[12345!]]]] ...
2
votes
0answers
147 views

PowersRepresentations Algorithm

I'm trying to understand the mathematics behind counting the number of representations of a positive integer by $n$ distinct $k$th powers, i.e. I would really like to know how to do the Mathematica ...