# Tagged Questions

Questions on the number-theoretic functionality of Mathematica.

4answers
764 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:
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 : ...
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 ...
6answers
768 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!]]]] ...
4answers
504 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 ...
2answers
288 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: ...
3answers
315 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 ...
6answers
395 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?
5answers
792 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.
1answer
306 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 ...
1answer
71 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. ...
1answer
239 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 ...
1answer
148 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, ...
1answer
94 views

### Implementing Remainder Tree

I want to implement Remainder Tree based on this. With the answers on SE I've come up with: ...
0answers
250 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 ...
3answers
142 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.
0answers
78 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 ...
1answer
282 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 ...
1answer
271 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 ...
3answers
214 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 ...
0answers
192 views

### Parallel PowerMod

Is there anyway to parallelize the PowerMod function? Here is my Left-To-Right modular exponentation: ...
2answers
189 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 ...
3answers
50 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?
3answers
247 views

### Generating pairs of additive and multiplicative factors for integers

Given an integer $n$, I want 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 came up ...
1answer
115 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 ...
1answer
55 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 ...
0answers
150 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 ...
0answers
85 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 ...
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