Questions tagged [number-theory]

Questions on the number-theoretic functionality of Mathematica.

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16
votes
4answers
871 views

Next highly composite number?

R language has this function 'nextn' (link) which computes the next highly composite number greater than a given one, which is used to find the optimal padding size for the subsequent FFT operation. ...
2
votes
1answer
469 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 ...
6
votes
1answer
278 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
514 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
1k 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 ...
4
votes
1answer
258 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. But ...
4
votes
1answer
776 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 ...
4
votes
1answer
476 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 ...
2
votes
3answers
264 views

Why do these two different zetas produce the same value? [closed]

Zeta[-13] == Zeta[-1] == -1/12 Why do these two different zetas produce the same value?
1
vote
1answer
170 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
4answers
1k 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 ...
11
votes
2answers
872 views

Modular arithmetic - efficiently calculating the remainders of factorials

When working on this question regarding the divisibility of the sum of factorials, I decided to write some code to test "small values" of the problem using the following code. ...
13
votes
1answer
443 views

What is the confidence limit on this convergence?

Bug introduced in 7.0 and fixed in 10.0.0 When I run this, Product[n^MoebiusMu[n],{n,1,Infinity}] I get $\frac{1}{4 \pi^{2}}$ Over on Math Overflow they ...
5
votes
2answers
391 views

How can this DivisorSigma code be made fast?

Since Project Euler problems are now fair game for questions I have a question of my own. A certain problem* states: For a positive integer n, let σ2(n) be the sum of the squares of its divisors. ...
11
votes
3answers
4k views

Easier program for period of Fibonacci sequence modulo p

For a little project I need to calculate the period of a Fibonacci sequence modulo p, for which p is a prime number. For example, the Fibonacci sequence modulo 19 would be: $$0, 1, 1, 2, 3, 5, 8, 13, ...
8
votes
3answers
369 views

How could I implement the equivalent of NextPrime

I would like to know what an implementation of the function NextPrime would look like if it were implemented in Mathematica's core language.
6
votes
1answer
315 views

Implementing Remainder Tree

I want to implement Remainder Tree based on this. With the answers on SE I've come up with: ...
1
vote
1answer
297 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 ...
14
votes
6answers
6k 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?
3
votes
2answers
544 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 it'...
15
votes
2answers
580 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: ...
13
votes
2answers
809 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
3answers
722 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 ...
2
votes
0answers
367 views

Parallel PowerMod

Is there anyway to parallelize the PowerMod function? Here is my Left-To-Right modular exponentation: ...
8
votes
1answer
252 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, $\...
10
votes
1answer
231 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 <...
2
votes
0answers
246 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 ...
37
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:
9
votes
0answers
848 views

Does Mathematica use the Elliptic Curve Method (ECM) in FactorInteger[]?

I'm not a mathematician, and I'm not even going to pretend that I understand anything of the ECM. But I know it can be a fast method for factorization. I benchmarked the factorization of $$...
6
votes
1answer
1k 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 ...
26
votes
1answer
4k views

Finding long strings of identical digits in transcendental numbers

Introduction Describing the three main streams of present-day mathematical philosophy (formalism, Platonism and intuitionism) in a well-known book, The Emperor's New Mind, R. Penrose says: ...it ...
12
votes
2answers
623 views

Ulam's Spiral with Oppermann's Diagonals (quarter-squares)

First we craft a function to return the quadrant boundary based on Oppermann's Conjecture a[n_] := (Mod[n, 2] + n^2 + 2 n)/4 Then we create a few lists ...
56
votes
2answers
5k views

What is so special about Prime?

When we try to evaluate Prime on big numbers (e.g. 10^13) we encounter the following issue: ...
14
votes
3answers
2k 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
405 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 ...
27
votes
7answers
3k 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!]]]] ...
21
votes
4answers
1k views

Why does Mathematica claim there is no even prime?

I wonder if this is a bug, or if I'm misunderstanding something: Exists[n, EvenQ[n] && PrimeQ[n]] // Resolve (* ==> False *) So if I interpret this ...
18
votes
6answers
3k views

Fastest square number test

What is the fastest possible square number test in Mathematica, both for machine size and big integers? I presume starting in version 8 the fastest will be a dedicated C LibraryLink function.