Questions on the number-theoretic functionality of Mathematica.

learn more… | top users | synonyms

-5
votes
1answer
52 views

Finding a seven-digit number with all of its prime factors less than 20? [on hold]

How can I find a seven-digit number with all of its prime factors less than 20? I have no clue how to do this.
4
votes
2answers
60 views

Goldbach Partition

I want to check the Goldbach conjecture for big number of $n$, but I don't know how to define this in Mathematica. There are my questions: Find a pair of primes $(p,q)$ for every even integer $n$, ...
0
votes
0answers
13 views

Show that 504 | (n^9 − n^3 ) for any integer n [migrated]

Not sure how to start this. I have equation of 504 is: 2 * 2 * 2 * 3 * 3 * 7, with $gcd(504)=7$
3
votes
1answer
49 views

A function about prime gaps

I want to define a $f$ function on Mathematica such as this. $f[k]$ gives the smallest $m$ holds $2k=Prime[m+1]-Prime[m]$. For example, $$f[1]=2$$ $$f[2]=4$$ $$f[3]=9$$ $$f[4]=24$$ How can i do ...
2
votes
1answer
103 views

One of the factors greater than $x$

Is there an easy way to tell Mathematica to find one of the prime factors of $n$ greater than $x$. For example, if $n=1299709\cdot 7919 \cdot 17$, is there a way to request a factor greater than ...
0
votes
1answer
224 views

Finite Field matrix rank calculation

How does one define a matrix over $\mathrm{GF}(p^r)$ in Mathematica in order to compute rank? I am working with $\mathrm{GF}(2)$?
0
votes
0answers
14 views

Euler's criterion [migrated]

An integer $n$ is a square modulo $p$ if there exists another integer $ x$ such that n $≡$ $x^2$ (mod $p$). Theorem 1 (Euler’s Criterion). : $1$. If $n$ is a square modulo $p$ then $n^2$ $≡$ $1$ (mod ...
1
vote
0answers
79 views

Truncate an infinite continued fraction at order 2000

I want to solve an equation which contains an infinite continued fraction F(n). Then I must (obviously) truncate this continued fraction at n=2000. The problem here is tha Mathematica does not ...
6
votes
2answers
856 views

Number of digits for factorial of 12345678987654321

A hard math problem What is the number of digits (IntegerLength) of the factorial of 12 345 678 987 654 321? The number of zeros at the end of this factorial was calculated and it is huge: exactly 3 ...
6
votes
1answer
83 views

Faster square test for integers

This question was asked already in Jan '12 and the most recent answer is from Oct '12, so it's several Mathematica versions out of date. What is a faster test for whether an integer is a perfect ...
0
votes
0answers
47 views

LatticeReduce question

Does LatticeReduce work with arbitrary precision arithmetic? That is, if I give it a linearly independent integer basis, but the integers are 40 decimal digits long ...
1
vote
0answers
35 views

Coppersmith's algorithm like Pari's zncoppersmith?

Is there some Mathematica package (or built-in that I missed) available, more or less equivalent to Pari's zncoppersmith function? Paraphrasing that source: given ...
4
votes
2answers
115 views

Finding vector of same direction with smallest integer coordinates

To determine Miller Indices of crystal lattice planes I would need a stable algorithm which determines the smallest set of integer coordinates of a vector which has same direction as a given vector ...
0
votes
1answer
69 views

Why does this function return the largest integer less than or equal to √n?

I've been asked this question by my teacher. The function I'm talking about is the following: ...
25
votes
2answers
2k views

Trying to Visualize a Collatz - 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 ...
48
votes
2answers
4k 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: ...
3
votes
1answer
79 views

Question about PrimeZetaP

The PrimeZetaP function appears to give results for complex s with real part > 0. Apparently, the analytic continuation is built ...
6
votes
2answers
162 views

Generating $\mathbb{Z}^*_n$

I'm using Mathematica to illustrate basic number theory concepts in a graduate cryptography class. To generate elements of the multiplicative group of integers modulo $n$, i.e. $\mathbb{Z}^*_n$, I can ...
2
votes
2answers
129 views

Can Mathematica return the first few terms of a sequence given the first few terms of a Dirichlet Generating Function?

For example: a=Sum[1/n^s,{n,1,6}]; Expand[a^2] returns a big mess. I want to see something like: 1/1^s + 2/2^s + 2/3^s + 3/4^s + 2/5^s + 4/6^s + ... + 1/36^s.
8
votes
2answers
221 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 ...
2
votes
2answers
162 views

Solving problem using recursive functions

Attached below is a question posed by the Canadian Mathematical Society, and I have my code and answer. Is there a better way of writing the code, and will the answer be different as a result? My ...
4
votes
1answer
90 views

Smallest integer that does NOT divide a given number

Divisible[n,m] yields True if n is divisible by m, and yields False if it is not. My query ...
4
votes
2answers
310 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 ...
-2
votes
2answers
102 views

How to efficiently find all sets of primes that add to a given number?

For large numbers the naive approach falls down: Select[Subsets[Range[1, 4]], PrimeQ@Total@# &] {{2},{3},{1,2},{1,4},{2,3},{3,4},{1,2,4}} ...
17
votes
7answers
1k views

Integers which are the sum of both two and three consecutive squares

This is a math problem I came across the other day: $365$ can be written as a sum of two and also three consecutive perfect squares: $$365=14^2+13^2=12^2+11^2+10^2$$ What is the next number with ...
7
votes
1answer
491 views

Calculating the density of nearest neighbours

I am trying to plot this which is a numerical simulation of the Montgomery-Odlyzko law for the nontrivial 1st $10^5$ zeros of the Riemann zeta function $ζ(s)$. The solid line is given by ...
3
votes
2answers
152 views

Pollard's Rho algorithm

I'm working in Mathematica and I'm trying to implement the [Pollard's Rho Algorithm for the Discrete Logartihm Problem][1].
2
votes
1answer
47 views

Non-integral common denominator

I have a list r = {114.49, 311.876, 538.704} whose elements are multiples of a non-integer value. I want to find the common denominator ...
1
vote
5answers
357 views

Prime factorization

I am trying to find a code that will output the prime factor decomposition of a number but for some reason I keep getting error messages. It is supposed to output the exponent of 2 and the odd factor. ...
1
vote
1answer
86 views

Can anyone re-produce this result related to the spectrum of Riemann Zeta using error term generated from MangoldtLambda?

All: I tried to reproduce the results from this page: How to plot the Riemann-Zeta zero spectrum The following is the code that was posted on above page: ...
2
votes
1answer
85 views

How can I plot the normalized distribution of the Riemann zeta zeros?

Given a list of eigenvalues or a list of Riemann zeta zeros, how can I plot this famous plot found here: On the page referred to, You need to click on "Programs", "The Riemann zeta function" and ...
13
votes
3answers
716 views

Proving (or at least 'being told by Mathematica') that Sqrt[2] is irrational?

I realize that Mathematica is not specifically an automated theorem prover. However, this article: http://www.wolfram.com/products/mathematica/newin6/content/EquationalTheoremProving/ Suggests that ...
13
votes
1answer
402 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 ...
1
vote
1answer
132 views

Find the maximum Z in {(X + Y)==Z} using all the digits 0-9 only once

II want to add two integers with different digits to get a third integer with different digits. At the end, all 10 digits have to be different. So there should be 10 digits in total. How you ...
0
votes
4answers
126 views
10
votes
1answer
201 views

Is there a PrimeQ whose accuracy guarantee you can adjust?

Say I have a list of a million integers each with a million digits, and I want a crude sieve to see which have a chance at being prime. Mathematica has a PrimeQ function, which appears to be slow ...
2
votes
0answers
65 views

Undocumented function SumOfSquaresReps

There is an interesting (and documented) number-theoretic function in MMA called PowersRepresentations[$n$, $k$, $p$]. It gives the distinct representations of the integer $n$ as a sum of $k$ ...
1
vote
1answer
182 views

How can I calculate all irreducible polynomials of 31 degree in $\mathbb Z_2[x]$?

How can I calculate all binary irreducible polynomials of degree 31? or how i calculate all irreducible $f$ in $\mathbb Z_2[x]$? (The irreducible polynomial in $\mathbb Z_2[x]$ and $\mathbb R$ are ...
3
votes
1answer
107 views

Range of summation in simple Plot seems off

I was trying to reproduce a picture in a book by Havil of the sum, $$s = \sum_{r=1}^{\infty}\frac{\mu(r)}{r}\left(Li(x^{\rho_k/r})+Li(x^{\rho_k*/r})\right) $$ using ...
9
votes
1answer
142 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 ...
3
votes
2answers
132 views

Symbolic multiplicative partitions

Let $p_n\#\equiv\prod_{k=1}^{n}p_k$ (primorial): p[n_] := Times @@ Prime[Range[n]] then the multiplicative partitions of $p_{1,2,3,4}\#$ are $$ \{\{2\}\},$$$$ ...
3
votes
1answer
108 views

High precision calculation of infinite product involving prime numbers

I'm recently studying some topics in analytic number theory and I have encountered results involving the infinite product $$C=\prod_{p}\left(1-\frac{1}{p(p+1)}\right)$$ where $p$ denotes calculating ...
0
votes
1answer
191 views

Prime number The Ulam spiral [closed]

I want a simple method of visualizing the prime numbers that reveals the apparent tendency of certain quadratic polynomials to generate unusually large numbers of primes. I was able to display the ...
21
votes
2answers
401 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 (...
11
votes
3answers
1k 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 ...
0
votes
1answer
82 views

solving quadratic and linear congruences with different modulus

We are able to solve the quadratic congruences $C^2 + Q^4 - 2\equiv mod 3072$ and $C - Q^2 - 2046\equiv mod 3072$ by entering ...
2
votes
1answer
139 views

What are the terms of the sequence generated by Zeta(3s)/Zeta(s)?

The LiouvilleLambda function has Dirichlet generating function of Zeta[2s]/Zeta[s]. I am curious about an analogous function with Dirichlet generating function of Zeta[3s]/Zeta[s]. Can Mathematica ...
6
votes
3answers
213 views

On a strange pattern of triangular numbers in Ulam's spiral

In this MSE post, user GeMir noticed that, where the green dots are the triangular numbers, $$T_n = \frac{n(n+1)}{2} = 1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,\dots$$ in the Ulam spiral ...
0
votes
0answers
38 views

How to print intermediate steps of simplifying a power formula? [duplicate]

To answer the question of proving Fibonacci sequence is periodic mod 5 without using induction., I came across Mathematica to prove $$F_{n}\equiv F_{n+20}\pmod 5$$ for all $n \geq 2$ I defined: ...
4
votes
1answer
105 views

Complex LogIntegral error

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