Questions on the number-theoretic functionality of Mathematica.

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1
vote
3answers
130 views

Finding primes that have certain property

Let S[p] denote the sum of digits of p. A prime p is said to be stubborn if none of ...
12
votes
3answers
485 views

Next highly composite number?

R language has this function 'nextn' (link) which computes the next highly composite number greater than a given one, which is used to find the optimal padding size for the subsequent FFT operation. ...
39
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 : ...
21
votes
4answers
799 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 ...
0
votes
1answer
34 views
3
votes
2answers
99 views

Generating 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 ...
1
vote
2answers
73 views

Arithmetic on algebraic numbers

I'd like to perform some elementary operations on algebraic numbers. ...
3
votes
6answers
356 views

Triangular numbers boolean function

I read the new book by Paul Wellin Programming in Mathematica. There is an exercise about triangular numbers. (The n-th triangular number is defined as the sum of ...
3
votes
3answers
166 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!
13
votes
1answer
369 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 ...
4
votes
1answer
154 views

Ruth-Aaron quadruple challenge

This a computational challenge, to find an efficient algorithm to discover a quadruple $(n,n+1,n+2,n+3)$ with the same sum of prime factors as described in the MO question, "Ruth-Aaron triples, etc." ...
11
votes
8answers
464 views

Find the minimum integer r such that $(10^r - 1)/37$ is an integer

I know Element[(10^r - 1)/37, Integers] tests the condition. So what is the command that gives me the minimum integer value r ...
14
votes
6answers
966 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 ...
6
votes
3answers
357 views

Better answer to Santa's riddle about sum of a number's divisors?

I was hoping to find an elegant solution to this riddle, using only a line or two of Mathematica: Santa Claus was telling one of his elves: "If I multiply the age of three of my reindeer, I get ...
2
votes
1answer
47 views

Factoring an ideal in a number field into prime ideals

I'd like to factor an ideal in a number field into prime ideals, exactly as in this example from the Sage documentation: ...
2
votes
3answers
129 views

Can you compute more terms in this sequence?

I am trying to identify a sequence related to the von Mangoldt function matrix. Since I believe/conjecture that the columns in the matrix have period lengths as in this sequence b: ...
0
votes
0answers
87 views

Solution of a set of quadratic congruences (Chinese remainder)

Edit: Please remove this question. I think there are mathematics error in what I am asking. You are welcome to edit the question if you can state the problem correctly. I have a set of quadratic ...
2
votes
1answer
82 views

One of the factors greater than $x$ [closed]

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 ...
1
vote
1answer
122 views

Mathematica spitting code back when using Resolve over a large range of interest

I've just started using Mathematica and have encountered my first issue. Below are two commands which only differ in the range of values I am asking Mathematica to check. The first works fine, but the ...
6
votes
2answers
253 views

A problem about fixed point iteration theory

Description Recently, I have been learning a couse called "Numerical Analysis". The fixed point iteration theory was introducted to solve the ...
4
votes
2answers
241 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 ...
6
votes
2answers
431 views

Counting the zeros in a factorial expansion [closed]

A number, as big as 1000! (! = factorial) is given. I need to find how many zeros are there in the number. I counted the no of terminal zeros by dividing 5, 25, 125, ... (untill fifth power < ...
1
vote
1answer
42 views

Q-Multinomial Coefficient [closed]

How can one compute the q-Multinomial Coefficient as a function of q,m and a list {n1,n2,n3,...} in Mathematica? See http://mathworld.wolfram.com/q-MultinomialCoefficient.html for the definition.
12
votes
1answer
378 views

the more effective method to find 21 digits armstrong number

In recreational number theory, a narcissistic number (also known as a pluperfect digital invariant (PPDI), an Armstrong number(after Michael F. Armstrong) or a plus perfect number) is a number that is ...
4
votes
4answers
271 views

Conveying density of 5-smooth (Hamming) numbers

A number is 5-smooth if its only prime factors are 2, 3 or 5. Example: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, … Interesting thing is that as they become larger ...
1
vote
5answers
256 views

Write a number as the product of its two largest divisors

For even n >= 10 && n <= 98 I want to write n as the product of its two largest divisors (excluding ...
2
votes
5answers
946 views

A question regarding 1 divided 243

Here is a problem due to Feynman. If you take 1 divided by 243 you get 0.004115226337 .... It goes a little cockeyed after 559 when you're carrying out the decimal expansion, but it soon straightens ...
6
votes
5answers
763 views

Write any positive integer as a sum of squares

With n = 17 I would like to get {4, 1} and with n = 999 {31, 6, 1, 1} so that, for example, ...
2
votes
1answer
103 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 $$ \{\{2\}\},$$$$ ...
36
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:
0
votes
2answers
113 views

Testing the Erdos square free conjecture

Can someone please answer these two parts of my homework assignment? Adding an explanation would be appreciated! a. Test the Erdos square free conjecture for $n <= 30000$. You should use ...
0
votes
0answers
219 views

Elliptic curve cryptography in Mathematica

I can find no resources for doing elliptic curve cryptography. I have used the finite field package, but I find it cumbersome and it does not seem to have any builtin methods for ECC. How can I get ...
0
votes
1answer
63 views

Square root of a value defined in a finite field?

I am trying to find the right way to compute the square root of a number defined in a finite field. For example, ...
3
votes
1answer
99 views
4
votes
0answers
90 views

Simplifying expressions involving Divisible

FullSimplify[ Divisible[p^2 - 1, 24] , Element[p, Primes] && p > 3] Should evaluate to True, but I get ...
2
votes
2answers
200 views

Checking if a number is a perfect power

I wanted to know how would I use Mathematica in order to check if the number is a perfect power I saw the algorithm but couldn't grasp it enough to implement it, so can anybody help?
4
votes
2answers
153 views

FactorInteger over UFDs

How can I factor 'integers' over other quadratic number fields (not just gaussian integers). For instance, how could I factor $7 = (3 + ω)(2 − ω)$ over Eisenstein integers ($ω = \frac{-1+ I ...
14
votes
4answers
2k views

Semi prime numbers

The high school textbook I am using has the example of semi-prime numbers. They wanted students to find (by "perspiration") all the semi-prime numbers less than $50$ (for a question on set theory). ...
1
vote
3answers
209 views

Generating a list of cubefree numbers

I am trying to generate a list of cubefree numbers (i.e. numbers when prime factorized contain no tripled factors) within a given range. Of DivisorSigma, ...
12
votes
6answers
2k 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?
1
vote
1answer
255 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 ...
8
votes
4answers
324 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 ...
3
votes
1answer
211 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 - ...
4
votes
1answer
338 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 ...
0
votes
0answers
106 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 ...
1
vote
3answers
164 views

Number theory: Problem involving rational numbers

Use RandomRat to test whether ((-1)^(1/Denominator[q]))^Numerator[q] is identical with ...
4
votes
4answers
553 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 ...
3
votes
3answers
198 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 ...
12
votes
5answers
655 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 ...
4
votes
1answer
408 views

Implementation of the Polynomial Chinese Remainder Theorem

I would like an implementation of the Chinese Remainder Theorem for polynomials in $\mathbb{Z}[x]$, that is, a function ...