Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [number-theory]

Questions on the number-theoretic functionality of Mathematica.

0
votes
1answer
25 views

running a loop to check for multiple congruences

I have the code for a loop to run. Right now it can check if a number is a square and at the same time check if it is congruent 0 mod 47. My questions is, how can I alter the code to see if it is ...
2
votes
1answer
140 views

Rewrite the power sum in terms of convolution

Let be an identity $$n^{2m+1}=\sum_{r=0}^{m}A_{m,r}\sum_{k=0}^{n-1}k^r(n-k)^r,$$ where $A_{m,r}$ are real coefficients, see A302971 for numerators and formula of $A_{m,r}$. Here we can notice that $\...
4
votes
1answer
838 views
0
votes
0answers
63 views

Most efficient way to define evil and odious numbers [duplicate]

A positive integer $n$ is defined to be evil if the number of ones in its binary expansion is even, otherwise it is odious. Now define the function $$ \begin{equation*} t(n)= \begin{cases} 1 \rm{ \...
2
votes
1answer
65 views

My version of PowerMod breaks down around 10^308

I have been trying to write a function that duplicates PowerMod[a, b, n], computing a^b mod n...
3
votes
1answer
162 views

Table of Chebyshev psi function

This should be easy, but for some reason I'm struggling. The Chebyshev psi function is given as the sum from 0 to x of the von ...
0
votes
0answers
44 views

28222149, a semiprime with amazing properties [migrated]

The semiprime $28222149$ is a semiprime $S=A*B$ (with $A=3$ and $B=9407383$) such that -$A.B$ -$B.A$ -$S.A$ -$S.B$ -$A.S$ -$S.A.B$ -$S.B.A$ -$A.S.B$ -$A.B.S$ -$B.S.A$ -$B.A.S$ are all ...
3
votes
1answer
54 views

List of invertible congruence classes

I am attempting to create a list of the invertible congruence classes $\bmod 120$. The code I have is ...
1
vote
2answers
56 views

Using a Do-loop to find divisors mod 13 [closed]

I want to check sum of divisors of i mod 13 fori = 1 to i = 20. I tried writing a Do-Print ...
1
vote
1answer
30 views

Mathematica code for computing the $p$-adic expansion of rational numbers

Does anyone know any Mathematica code for computing the $p$-adic expansion of rational numbers? I.e. given a rational number $a/b,~a,b\in \mathbb{Z}$ and a prime number $p$, then compute the $p$-adic ...
6
votes
1answer
63 views

Computing the seven roots of a polynomial

This question was originally asked by @fsrong70 six months ago. The OP deleted it shortly after posting and has not returned to this site since. I had just figured it out when it was deleted. I ...
4
votes
0answers
84 views

Not comprehending PowerMod

Bug introduced in 10.3 or earlier and persisting through 11.3.0 or later The bug is not present in 10.0. By definition, PowerMod[a, 1/r, m] finds a modular rth root of a mod m. Here's a pair of ...
3
votes
2answers
793 views

Determining if a number is divisible by 1000 [closed]

I have a number such as: a = 875952; And I want to find if it is divisible by 1000. Is there a concise way of doing that?
4
votes
3answers
347 views

Generate numbers relatively prime with a given number

I am interested in a function such that f[m, i] = n where m, n are positive integers and n ...
0
votes
0answers
69 views

Find the maximum of a function involved with Floor function

My function is $$ f(H,p) = \left\lfloor \dfrac{\lfloor H/p\rfloor + 3 - \sqrt{(\lfloor H/p\rfloor + 1)^2 - 4H}}{2} \right\rfloor $$ The constraints are $ H \geq p(4p-1) $, $ p $ is prime although ...
0
votes
1answer
65 views

What is the best build-in function to pick or select given element in a huge list

im looking for a best and efficient function works like a search engine it takes for example m=5+6I ,then it goes searching in the list V={1,1+I,2+3I,...} until catch it . My Dr said to use "Select[]"...
7
votes
2answers
287 views

Efficient code for minimum integer with given number of factors

I'm seeking an efficient implementation of the number-theoretic function giving the smallest integer $n$ that has exactly $k$ factors (not necessarily prime): ...
6
votes
2answers
146 views

Finding $x$, the exponent of a $2^x$ when we need an specific output

Say we need a program Findx[n_Integer, m_Integer] where n is an integer from 1 to 9 and m is an integer from 1 to 1000. The output of the program is a number x, which is the exponent of 2 that ...
-5
votes
1answer
143 views

Solving Diophantine equations 5 [closed]

Given a positive integers x,y,m would like to be able to find integer solutions z from Diophantine equation x^2-y^2 = m*z in Z. ---The proof,is divided in 4 sections---... x=2*k+1 ,y=2*h+1 with ...
1
vote
0answers
47 views

Problem in Counting the Number of Divisors with a Function [closed]

I want to create a function in mathematica which returns the number of divisors of a parameter x. So, I created the following function: d[x_] = Length[Divisors[x]]; And when I evaluate: d[1] d[2] ...
1
vote
1answer
24 views

Finding smallest domain within which variables can satisfy inequality

Given an arbitrary number of variables $\epsilon_i$ that can be picked from a domain $[0, W]$ and some inequality relation between all the variables $G(\epsilon_1, \epsilon_2, ...)$, is there some way ...
8
votes
5answers
354 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?
3
votes
0answers
95 views

How to find, load and use a legacy Mathematica package

Question: How to find, load and use a legacy Mathematica package from a previous Mathematica version? What is the preferred approach and process? Context: I am working through a set of examples in ...
1
vote
1answer
39 views

FunctionDomain in the Reals numbers [closed]

Why don't I obtain that $x$ belongs to the Reals? ...
-2
votes
2answers
86 views

Simultaneous equation [closed]

How to solve Simultaneous equation in Mathematica: x^2 - (a^2 - 1)y^2 = 1 and y^2 - pz^2 = 1 where ...
1
vote
1answer
84 views

Number-theoretic function using Table

I'm trying to make a function to calculate the Log sum of primes over a limited range $1/2n$ to $n$ or Chebychev theta function over limited range $1/2n$ to $n$. This will be used only for even ...
7
votes
4answers
294 views

Find the order $m$ of a matrix ${\bf A}$ such that ${\bf A}^m= {\bf 1}$

I have a square matrix ${\bf A}$ defined over the field $\mathbb Z_2$ and I want to find its order such that ${\bf A}^m=1$. I tried using ...
2
votes
1answer
41 views

How to define a function the Dirichlet L-function $L(s,\overline{\chi(5,2)})$ in Mathematica?

In Mathematica: The Dirichlet L-function with character $\chi(5,2)$, $L(s,\chi(5,2))$, is expressed as DirichletL[5,2,s] Let $\overline{\chi(5,2)}$ be the complex ...
11
votes
3answers
1k views

François Viète's approximation to π

How do I program the approximation to π devised by François Viète, which is given by 2 * 2/Sqrt[2] * 2/Sqrt[2 + Sqrt[2]] * 2/Sqrt[2 + Sqrt[2 + Sqrt[2]]] * ... ...
0
votes
0answers
42 views
0
votes
2answers
92 views

Code for finding $a$ and $b$ such that $a b = 1 \mod 4$ [closed]

I need to find $a$ and $b$ such that $a b = 1 \mod 4$? I do not know how to write the code. Could someone help me?
3
votes
1answer
75 views

Generating numbers palindromic in two number bases

The purpose of the code below is to generate numbers that are $2d+1$ digit palindromes in number base $b+1$, and are also palindromic in number base $b+3$, where: The ...
0
votes
1answer
165 views

Easy number theory problem

$p$ is an odd prime number,$S = \{x \mid 1 \leq x \leq 2p, x \in \mathbb{Z} \}$, $A$ is a subset of $S$, satisfying $\operatorname{card}(A) = p$ $\sum\limits_{x\in A} x \equiv 0 \pmod p$...
0
votes
2answers
121 views

Iterative calculation of a number-theoretical constant [closed]

Recently the decimal expansion of a number theoretic constant was searched for, which is the analog of the Landau-Ramanujan constant in a certain context. The constant starts 0.638909... It can be ...
6
votes
1answer
96 views

Finding minimum x such that Mod[3^x, m] == 1 for m not multiple of 3 [closed]

I would like to find smallest x value for each m value such that Mod[3^x, m] == 1, where m is not multiple of 3. Here is my ...
2
votes
0answers
63 views

Listing divisors of a number [closed]

I created a list with all divisors of 18000: list1 = Divisors[18000] I did an analysis to identify which of these numbers are divisible by $15$, but the result I ...
5
votes
5answers
206 views

Question about how to speed up Mathematica code

When looking at the Minimal Goldbach prime partition point {p,q} for each n; where n=10^i and i = 2,3,4,...,10; I notice that these points reside in an interval with center n/2 and radius 250: ...
4
votes
1answer
177 views

solid partitions generator

According to Wikipedia, a solid partition of $n$ is a three-dimensional array $n_{i,j,k}$ of non-negative integers (with indices $i,j,k \geq 1$) such that $$\sum_{i,j,k} n_{i,j,k}=n$$ and $$n_{i+1,j,k}...
3
votes
1answer
111 views

How can I define $i^2=j^2=k^2=-1$ in Mathematica?

First, I want to define the identities as $i^2=j^2=k^2=-1$, $ij=k=-ji$, $jk=i=-kj$, $ki=j=-ik$. And then I want to use these identities in my sequence $Q_n = F_n + iF_{n+1} + j F_{n+2} + k F_{...
9
votes
4answers
369 views

First two n such that $1355297$ divides$10^{6n+5}-54n-46$ for $n>0$

I have problem solving this equation, smallest n such that $1355297$ divides $10^{6n+5}-54n-46$. I tried everything using my scientific calculator, but I never got the correct results(!).and finally I ...
0
votes
1answer
55 views

function that generates a list of all plane partitions of a given dimension

Is there a function in Mathematica that generates a list of all plane partitions of a certain dimension $n$? This paper describes the algorithm, but I still find it a bit tricky to do it myself.
0
votes
1answer
87 views

Understanding the question

Let n be the integer shown below: ...
0
votes
1answer
103 views

Generate a prime number satisfying a condition

Building on the same context of this question Is it possible to enforce constraints on the properties of the generated prime? For example, I'm trying to find a prime $p$ in the range specified in the ...
3
votes
3answers
636 views

Solving a system of diophantine equations from a mathematical competition

There is a problem from the $66th$ Putnam Mathematical Competition, $B2$ Find all positive integers $n,k_1,k_2,...,k_n$ such that $$k_1+\cdots+k_n=5n-4$$ and $$\frac{1}{k_1}+\cdots+\frac{1}{...
15
votes
1answer
546 views

InverseTotient[ ]?

Maple has a function InverseTotient( c ), which returns all those natural numbers $n$ whose Euler totient function $\phi( n ) = c$. Is there an equivalent inverse ...
8
votes
3answers
301 views

How to represent integers using Egyptian fractions?

Let $N(n)$ be a set of integers, which can be presented using first $n$ Egyptian fractions: $$ N(n):=\{m\in\mathbb{Z}:\ \ m=\sum_{i=1}^n\frac{\epsilon_i}{i},\ \epsilon_i=0\ \text{or}\ 1\} $$ I want ...
1
vote
1answer
83 views

Solving a modular equation with a large modulus [duplicate]

I am having trouble solving this equation: $17^n\equiv 1 \mod 640826899755722841329632946651705039010285107743541637238400825304943424424715705661705191669759$ for the smallest integer $n > 0$ I ...
3
votes
1answer
174 views

Generation of Step Numbers

I am working on Project Euler 178 but got stuck in trying to optimize my code. The following text comes from the problem: Consider the number $45656$. It can be seen that each pair of ...
1
vote
1answer
117 views

Calculate 40 digits of the MRB constant

MRB constant is the upper limit point of the following sequence $$s_n=\sum_{k=1}^{n} (-1)^k k^{\frac{1}{k}}$$ $MRB=\color{blue}{0.1878596}...$ I tried to calculate first few digits: ...
0
votes
0answers
50 views

Question about how to use NestWhileList

I start with: n = 2228; m = n/2; PreviousPrime[n_] := NextPrime[n, -1] I use NestList to build the following list: ...