# Tagged Questions

Questions on the number-theoretic functionality of Mathematica.

2answers
95 views

### Need help with code for number theory problem

I'm completely new to Mathematica (used previously only for very simple cases). I need to write a quite complex function. The function must do the following: Input consists of two numbers: a and b. ...
1answer
210 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 $2345671$...
5answers
362 views

### On finding all the positive integral solutions of $x^2+y^2=z^2+1$

I am a new to Mathematica. My goal is to find many (if not all) positive integer solutions to the equation: $x^2+y^2=z^2+1$ using Mathematica. However the problem is that I can only find a ...
3answers
753 views

### How can FactorInteger be so slow if PrimeQ is fast?

My 8th grade son had a homework problem to find a prime factor of $99!-1$. I thought to be clever/lazy and used FactorInteger[99!-1], but it takes forever. ...
3answers
137 views

### Expressing a series formula

I want to generate a series of the following kind in Mathematica: $\quad \quad a(n+1) = a(n) + ({\rm prime}(n+1) - 1)/2 \quad \mbox{for odd primes},$ so that the resultant series is ...
3answers
164 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 ...
3answers
674 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. ...
4answers
910 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
95 views

### Arithmetic on algebraic numbers

I'd like to perform some elementary operations on algebraic numbers. ...
6answers
574 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 ...
3answers
229 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!
1answer
175 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." ...
8answers
608 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 ...
3answers
490 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 ...
1answer
136 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: ...
3answers
142 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: ...
0answers
109 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 ...
1answer
131 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 ...
2answers
395 views

3answers
233 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, ...
6answers
3k 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?
1answer
271 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 ...
4answers
359 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 ...
1answer
378 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}...
1answer
550 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 ...
3answers
188 views

### Number theory: Problem involving rational numbers

Use RandomRat to test whether ((-1)^(1/Denominator[q]))^Numerator[q] is identical with (-1)^q...
3answers
227 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 ...
5answers
814 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 ...
1answer
612 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 ...
3answers
305 views

### Find integer values of p such that $(2^p - (2^2)(3^2))/ (3^3)$ is an integer

Find integer values of p such that $(2^p - (2^2)(3^2))/ (3^3)$ is an integer.
1answer
490 views

### random number visualization without generator (Spectral Test) [closed]

Does anyone have codes for using only numbers to make this kind of random-number visualization and not requiring a number generator? http://demonstrations.wolfram.com/...
1answer
338 views

### Expressing large numbers in index form

I have a quick question. Is there anyway of expressing a large number as a power of another number in Mathematica? By this, I mean for example, $1237940039285380274899124224 = 512^{10}$. Is there a ...
3answers
931 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 ...
1answer
203 views

### How can I program the RiemannR function using the LogIntegral command?

I would like to program the RiemannR function using the LogIntegral command because I would like to later experiment with a ...
1answer
257 views

### Hermite Normal Form in “columns” convention

After describing the Hermite Normal Form (HNF), MathWorld explains: The Hermite normal form for integer matrices is implemented in Mathematica as ...
1answer
939 views

### Function to Determine Lucky Numbers

Given a list of the form {1, 3, 5, 7, ...}, the lucky numbers are obtained by looking at the first list element after 1 (so 3 in this case), and deleting all list ...
3answers
602 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 ...