# Tagged Questions

Questions on the number-theoretic functionality of Mathematica.

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The PrimeZetaP function appears to give results for complex s with real part > 0. Apparently, the analytic continuation is built ...
152 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 ...
77 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 ...
133 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].
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### 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 ...
56 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 ...
345 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. ...
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### 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: ...
130 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 ...
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### 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 $n$?
100 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}} ...
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### 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 $k$ ...
106 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 ...
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### 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 ...
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### 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 (...
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### 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: ...
194 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 ...
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### Write a function pollard[n, B] that tries to factor an integer n, using Pollard's p − 1 method with at most B iterations [closed]

This is what I got but it seems it's not working. When I test it, it just goes through and nothing gets returned. Is there something I'm missing? ...
81 views

### Iterative Tree Plot for the Sum of an Integer's Digits Squared

I am trying to make a graph that depicts integers<100 mapping to the sum of their digits squared. I can do this for one iteration, but I don't know how to do it for more than one, or until the new ...
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### Dirichlet series expansion

Is there a way to get Dirichlet series expansions in Mathematica? For example, I would want DirichletSeries[Zeta[s], {s, 4}] to return ...
2k views

### Trying to Visualize a Collatz - The Collatz conjecture

So I'm new in this and learning--- and 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 ...
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### 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 ...
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### Find all “chains” in the poset of divisors

I want to input a set of divisors of an integer $n$ and return all subsets of these divisors ${d_1,d_2,...d_k=n}$ such that $d_1$ divides $d_2$, $d_2$ divides $d_3$, ... and $d_(k-1)$ divides $d_k$. I ...
254 views

### Permuted Prime Numbers

How can I produce all 3-digit and 4-digit prime numbers [100-9999] in which, all permutations of all digits produce again a prime number, such as 311, 131, 113, ...
230 views

### Possible improvements to this Syracuse (3x+1)/2 graph?

This algorithm produces the Syracuse disjoint tree graph without any duplicates. No need for Union, For, and ...
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### List of prime powers

I have a list of not necessarily distinct prime powers. For example: {2,3,4,25,2,3}. I want to combine (multiply) the highest prime powers for each prime. In this case 25*3*4 = 300 since 25 is the ...
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### How to further accelerate arithmetic with Fermat Pseudoprime and Fibonacci number

I've been working on this all night, and I have made this go pretty fast, compared to my first iteration of the program, but now I'm out of ideas. I'm trying to write a program to test (by good ...
202 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 ...
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### 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. ...
153 views

### Using the Baby-Step Giant-Step algorithm

Here is a concept I am working through: As part of an attack on an El-Gamal cipher, solving the discrete logarithm problem $$10^x = 532107 \;\, {\rm mod} \;\, 1313839.$$ Using the ...
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### 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 ...
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### 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 ...
302 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 ...
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### 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 ...
131 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 ...
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### How to “de rationalize” a number? [closed]

How can 1/2 be represented as 0.5? Thanks!
133 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 ...
187 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 $\mathrm{GF}(2)$?
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### 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 ...
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### Arithmetic on algebraic numbers

I'd like to perform some elementary operations on algebraic numbers. ...
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### 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!
617 views

### Calculating weird numbers

A weird number is a number such that the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of these divisors sums to to ...
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### 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.
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### 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. ...
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### 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 ...
556 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 ...
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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." ...
408 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 ...
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### 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: ...
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### 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: ...