Questions on the number-theoretic functionality of Mathematica.
2
votes
1answer
93 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 ...
13
votes
3answers
347 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
...
4
votes
1answer
76 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
59 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
160 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
269 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
120 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. ...
9
votes
2answers
349 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 ...
3
votes
1answer
95 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 ...
29
votes
2answers
3k 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 :
...
2
votes
3answers
137 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?
2
votes
1answer
89 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
119 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 ...
3
votes
3answers
234 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 ...
7
votes
1answer
72 views
ToNumberField won't recognize Root[…] as explicit algebraic number
In Mathematica 9.0.1, it appears that ToNumberField will not always recognize a Root object as an explicit algebraic number.
...
2
votes
2answers
194 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 ...
10
votes
6answers
431 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?
4
votes
1answer
276 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 ...
7
votes
1answer
247 views
What is the confidence limit on this convergence?
When I run this,
Product[n^MoebiusMu[n],{n,1,Infinity}]
I get $\frac{1}{4 \pi^{2}}$
Over on Math Overflow they are saying it shouldn't happen. So, how do ...
7
votes
1answer
151 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, ...
5
votes
3answers
144 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
98 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
0answers
154 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 ...
13
votes
2answers
301 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
194 views
Parallel PowerMod
Is there anyway to parallelize the PowerMod function?
Here is my Left-To-Right modular exponentation:
...
6
votes
1answer
305 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 ...
5
votes
0answers
81 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 ...
9
votes
5answers
827 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.
20
votes
6answers
834 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
86 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 ...
27
votes
4answers
773 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:
7
votes
0answers
264 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 ...
17
votes
4answers
515 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 ...
21
votes
1answer
2k 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 ...
