Questions tagged [number-theory]

Questions on the number-theoretic functionality of Mathematica.

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58
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: ...
24
votes
8answers
4k 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.
13
votes
5answers
1k 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 ...
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 ...
14
votes
6answers
7k 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?
36
votes
2answers
9k views

Trying to visualize the Collatz conjecture

I happen to have this collatz collatz[x_, y_] := If[x == 3*y || x == 2*y + 1 || y == 3*x || y == 2*x + 2, 2, 0] So i want a visual 3D adjacency graph of my ...
27
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 ...
4
votes
3answers
550 views

Multiplicative partition function

I am trying to create a multiplicative partition function that would generate something like ...
7
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 ...
24
votes
3answers
604 views

Speeding up the built-in Rudin-Shapiro and Thue-Morse sequence functions

Version 10.2 introduced two well-studied sequences as functions: the (Golay-)Rudin-Shapiro sequence (RudinShapiro[]) and the (Prouhet-)Thue-Morse sequence (...
8
votes
4answers
886 views

Goldbach Partition

I want to check the Goldbach conjecture for a big number of $n$, but I don't know how to define this in Mathematica. These are my questions: Find a pair of primes $(p,q)$ for every even integer $n$, ...
4
votes
1answer
520 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
275 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?
4
votes
3answers
771 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 ...
8
votes
2answers
1k views

Plotting the sum of two points on an elliptic curve

I am doing an experiment to prove the associativity of the addition of points on an elliptic curve. So far, I have produced a code which allows me to move points on my curve. To find their sum, I ...
7
votes
2answers
979 views

Approximation to the prime counting function

Is there a function similar to PrimePi that gives approximate value for large numbers? In particular, I need a reasonably good approximation for $\pi(x)$, where $x\...
38
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:
15
votes
2answers
601 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: ...
11
votes
2answers
1k views

Finding Ramanujan's taxicab numbers

How to find Hardy-Ramanujan Numbers by using Mathematica? Definition: Taxicab number is defined as the smallest number that can be expressed as a sum of two positive cubes in $n$ distinct ways. ...
9
votes
2answers
586 views

Number of divisors visualized with the QPochhammer function, how to improve performance of code?

I have this code that is originally Jeffrey Stopple's code for the Riemann zeta function in the complex plane. Because I discovered yesterday that the number of divisors can be generated with the $q$-...
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!]]]] ...
13
votes
2answers
877 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 ...
9
votes
2answers
785 views

Visualizing the primes with the Riemann Zeta function

I am trying to plot the identity seen here, namely that if we define: $$\psi _{0}(x)={\frac 12}\left(\sum _{{n\leq x}}\Lambda (n)+\sum _{{n<x}}\Lambda (n)\right)$$ Then, it equal to the following,...
8
votes
5answers
630 views

Does Mathematica have a twin prime equivalent of `PrimePi`?

Well, it's all there in the title! I'd like to be able to plot the number of twin primes =<x. Is there an inbuilt function that can do this?
4
votes
3answers
330 views

Code for (a,b) with gcd(a,b)=1?

I am trying to make a big table that includes all ordered pairs (a,b) with a (1,2) (1,3) (2,3) (1,4) (3,4) (1,5) (2,5) (3,5) (4,5) (1,6) (5,6) ... Any ideas? Thanks!
2
votes
1answer
1k views

Factoring large integers with the Pollard p-1 method

I am trying to use the Pollard $p-1$ method to find the factors of a large integer. Here is the problem: An RSA-type cipher is based on the integer $n = 140016480344628383$ and exponent $2345671$....
1
vote
4answers
500 views

How can I write the natural numbers less than $n$ that are coprime to $n$? [duplicate]

How can I write the natural numbers less than $n$ that are coprime to $n$?
10
votes
4answers
438 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 ...
7
votes
5answers
2k 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 ...
6
votes
1answer
197 views

Complex LogIntegral error

Going through Derbyshire's Prime Obsession & trying to take LogIntegral of 20^ZetaZero[1] & comes up with a value of <...
0
votes
1answer
114 views

Writing a program that finds for what $(x,y)$ a function gives a perfect square number

The overal question I am trying to answer is: For what $(x,y)$, which are positive integers, is the following number a perfect square number? $$9 \left(x^3 (y-2)^2+3 x^2 (y-2)-2 x (y-45) (y-2)+7 (y-1)^...
5
votes
1answer
566 views

Solving a Diophantine equation with a large solution

I am trying to solve the following Diophantine equation with Mathematica: $\frac{x}{y+z}+\frac{z}{x+y}+\frac{y}{x+z}=4$ It is known that there are three positive numbers that satisfy the equation ...
2
votes
0answers
377 views

Parallel PowerMod

Is there anyway to parallelize the PowerMod function? Here is my Left-To-Right modular exponentation: ...
1
vote
3answers
231 views

Number theory: Problem involving rational numbers

Use RandomRat to test whether ((-1)^(1/Denominator[q]))^Numerator[q] is identical with (-1)^q...
1
vote
1answer
302 views

How to solve equations in Gaussian integers modulo p

How to solve a complex equation of the form: $$z^n \equiv i \pmod p$$ where $z$ is a Gaussian integer, $i$ is the imaginary number, $n, p \in \mathbb{Z}^+$ and $p$ is prime. I am dealing with quite ...
0
votes
2answers
227 views

Powers of prime factors of a positive integer $n$ in “Mathematica”?

I would like to find the powers of a prime in the unique prime factorization of an $n$. I want a function $f[n,p]$ such that $n,p$ are given and I need to know what the power of $p$ is. For instance ...
19
votes
4answers
3k views

Find the 5566th digit after the decimal point of 7/101

I want to find the 5566th digit after the decimal point of 7/101. I input the following code into Mathematica 11: Mod[IntegerPart[7/101*10^5566], 10] The output ...
17
votes
1answer
1k views

Visualisation of the field of algebraic numbers in the complex plane

Hot to plot the field of algebraic numbers in the complex plane? In this picture, the color of a point indicates the degree of the polynomial of which it’s a root: red = rational numbers ...
13
votes
3answers
4k views

Has Mathematica a function to compute the Smith Normal Form?

The Smith normal form is a matrix that can be calculated for any matrix (not necessarily square) with integer entries. See Wikipedia for a more elaborate description. Has Mathematica a function to ...
7
votes
3answers
314 views

Calculating the integral points of an elliptic curve

I asked this question on Math stachexchange. The question I have is: Can I use Mathematica to find the (integral points on the following elliptic curve) or can I find when the number $\text{n}$ is a ...
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, ...
6
votes
1answer
438 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 ...
5
votes
1answer
1k 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 ...
3
votes
2answers
144 views

Using the solve function for big numbers, getting a failure now

When I try to solve: Solve[y^2==441+48*x*(1+x)(-13+16*x)&&1100*10^9<=y<=1200*10^9&&x>=2,{y,x},Integers] My code runs for 169 seconds and ...
3
votes
1answer
591 views

How does Mathematica compute how to write integers as the sum of k non-negative pth integer powers so quickly?

"PowersRepresentations[n,k,p] gives the distinct representations of the integer n as a sum of ...
13
votes
1answer
469 views

SquaresR memory leak?

I have tried the following code in Mathematica 11.0.1.0 on my MacBook: ...
5
votes
1answer
1k 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 - \tfrac{1}{k_n}...
5
votes
2answers
1k views

Recursive Euclidean algorithm in Mathematica

Can anyone explain to me how do I use a recursion, if I don't know the limit? For example, I need the remainder $r$ of the Euclidean algorithm for $\gcd(a,b)$ which equals $0$. I figured out that the ...
3
votes
2answers
218 views

solve for two variables for each n related to Collatz conjecture

For this code, for each x I would like to solve for all value ranges for c1 and c2 in a bounded range ie c1 and c2 in the range of real numbers +-100 for c1 and c2 for each x, which combined give "...
3
votes
3answers
256 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 ...