# Questions tagged [number-theory]

Questions on the number-theoretic functionality of Mathematica.

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439 views

### All solutions that satisfy $x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}-x_{5}^{5}=0$

I want to find a combination that satisfies all the solutions of the following formula. $$x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}-x_{5}^{5}=0$$ $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$, and $x_{5}$ are ...
186 views

### How to make FactorInteger iterative?

I want to factorize big numbers like 10^100. FactorInteger with no Automatic option can take a lot of time and as I know there ...
37 views

### Manipulating Dirichlet characters and L functions

I read some basic knowledge about characters and L functions, and would like to play around with them in MMA. I tried to do the following things, but ending in minor success. (MMA notation used) ...
102 views

### How to make a function that returns all super distinct partitions?

I am working on distinct partitions. I recently created a function StrictIntegerPartitions. This is from the book Integer Partitions by George E. Andrews at Pennsylvania State University and Kimmo ...
686 views

### Design a function that gives all strict partitions of an integer

A strict partition of an integer has all distinct parts. There are no duplicates like 1 in {5,3,1,1} for 10. All the parts are unique. Sylvester created a bijection between the partitions of an ...
1k views

### Solve Olympiad Problem with Mma

Find all integers $k\le100$, so that there exists an integer $n$, satisfying $k\mid3n^6+26n^4+33n^2+1.$ By number theory knowledge it suffices to check $n\in[1,k]$, but we'll do $[1,100]$ for ...
42 views

### Using "ToNumberField" as opposed to equations over the integers

Let D>1 be a square-free rational integer, and write \Q for the rationals. I am trying to determine the (non-)membership of ...
1 vote
94 views

### Why do some functions, or at least, PowersRepresentations, run far faster on subsequent calls? What determines this behaviour?

Say I run the following: PowersRepresentations[4782969,4,2] and it takes about 2 minutes. If I call it again it takes only about 0.0005 seconds. What determines ...
60 views

### Does applying Reduce result in an equivalence or a oneway implication or both? [closed]

If I type something like this into Mathematica: ...
1 vote
80 views

### Select primes from their Zeckendorf representation

I'm working with the Zeckendorf representation of prime numbers. I'm using ResourceFunction["ZeckendorfRepresentation"][Prime[n]] and I would like to select from all the results, the ones ...
1 vote
63 views

### Representing a number in r0 + r1 E + r2 E^2 form

Let E be the base of natural logarithm 2.71... A Sequence S[n] is believed to converge to a ...
316 views

### How to determine the unique combination of digits satisfying a given relation?

Is there a method to determine the unique combination of numbers a, b, c and d which satisfy the relation below, and which yields the output with the numbers in the given order. Example for 2023 is ...
626 views

### Can I use NextPrime[n] up to n=10^14?

I would like to perform computations with primes up to $n=10^{14}$. To do so, I would like to go through all primes, from $2$ to $10^{14}$ and perform some calculation on each prime. I saw that one ...
630 views

### How to ask Mathematica to compute the given sum of the differences of the numbers of the given two sets?

I have two sets of real numbers, say, set1= {b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11} ...
1 vote
135 views

### A square board of size n*n [closed]

I would like to Randomly generate n^2 natural numbers in an interval from 1 to n. Then consider placing each number on the cell with the same number (you can imagine the board numbered naturally, row ...
178 views

### Writing the number '80668227' as a sums of 4 & 5 cubes

I need to write the number '80668227' as a sum of 4 & 5 cubes. I tried this code PowersRepresentations[80668227, 4, 3] in Mathematica but the above code is ...
60 views

### Dedekind Zeta Function in Mathematica (at least for quadratic number field)

Does there exist some way to use Mathematica to compute the Dedekind Zeta function for an arbitrary algebraic number field? Or does there exist some package to do this? I am actually only interested ...
1k views

### Finding least n such that n^2 + 23 is divisible by large powers of 2

Lets say that we want to find the least n such that n^2+23 is divisible by 2^100. We can compute this in one line using the Pari/GP language: ...
286 views

### Find all sets whose index is divisible by the elements

Suppose I have such a set $S=\{3,4,5,...n+2\}, n\in \mathbb{Z}_{>\, 0}$, $a_n$ is any permutation of $S$ such that $n|a_n$, I want to know how many such $a_n$. The easy but inefficient way to do ...
98 views

### How to Make a list or table containing the product of every three digit integer in mathematica

I am trying to learn the mathematica language. And it was suggested to me to start by doing the Project Euler problems. I am currently working on Problem # 4: A palindromic number reads the same both ...
92 views

### Test if the decimal digits of $n$ appear $n$ times in the decimal representation of $n!$

I want to find numbers $n$ for which the decimal digits of $n$ appear $n$ times in the decimal representation of $n!$. For instance, $n\in\left\{0, 1, 1170, 1528, 9877, 9886, 9897, 11535\right\}$ are ...
169 views

### System of equations

I am trying to find ALL or a list of possible values for x satisfying: x MOD 9 ==2 AND (x+631) MOD 9 = 3 Using Mathematica. can anyone help? 2 is not the only answer 497 is another one since 497 mod ...
109 views

105 views

### Check certain expression using a while loop to run through all posibilities in a range

Well, I have written the following code (using the fast square root test found in this answer): ...
273 views

### Longest arrangement of n-digit square numbers s.t. last digit equals first digit of next

I have this problem: consider all the square numbers with exactly n digits, I want to arrange them such that the last digit of a square is equal to first digit of the next square and find the longest ...
1k views

### The integer ababab (a,b>0) is always divisible by 7, without remainder

The integer $ababab$ $(a>0,b>0)$ is always divisible by $7$, without rest. I tried to prove this by: ...
195 views

### How do I maximize the following?

I want to maximize $${T(\epsilon)=\max\left\{\log_{(1/y)}\left(|\sqrt{2}-x/y|\right):x,y\in\mathbb{N}, |\sqrt{2}-x/y|<\epsilon\right\}}$$ Edit: I changed the inequality, it was supposed to be ...
105 views

### Divisibility problem! I cannot find a proof. Help would be appreciated! [closed]

It appears that IntegerQ[(2 k - 2^k)/(2 k + 1)] and IntegerQ[(2^k + 1)/(2 k + 1)] have the same k values. Proof?
144 views

### Faster PowersRepresentation using IntegerPartitions

Earlier I posted a question about taking fast integer square roots of known integer perfect squares. The reason this came up is I was trying to find a faster way of mimicking the PowersRepresentations[...
140 views

### Mathematica code for q-Stirling numbers

In the paper [A new $q$-Analog of Stirling Numbers],(https://hal.archives-ouvertes.fr/hal-01372920/document)[PDF] J. Cigler defined $q$-Stirling numbers of the second kind as the following: He ...
60 views

### Check if a fraction leads to an integer, choose $r$ and run trough $0\le a\le n$ and go to $r+1$

Well, I am trying to do the following: I have two functions $y(r,a)$ and $z(r,a)$ and I want to check of the division of both those functions leads to an integer. In order to do that I want to choose ...
116 views

### Finding IntegerPartitions with no zero and no duplicates

Well, I am trying to execute the following code: ...
49 views

### Alternative for PowerRepresentations[n, x, 1]

I am looking for numbers $n$ that can be written as the sum of other numbers. So: PowersRepresentations[n, x, 1]. But is there a faster way of finding these numbers?...
67 views

### Fourier expansion of Dedekind eta function with rational arguments

I want to be able to compute the Fourier series expansion in q=Exp[2πiz] for DedekindEta[(az+b)/(cz+d)] with integer a,b,c,d, but not requiring ad-bc=1. Just replacing z in terms of q and doing ...
327 views

### Checking if a number is right sorted

I have a number $n$ such that the digits of $n$ are strictly increasing to the left except for the first digit. So for example when $n=51369$ fits the bill because: $$1<3<6<9\tag1$$ Is there ...
168 views

### Calculate the Leyland number index of a given Leyland (x,y) pair

A Leyland number L(x,y) is x^y + y^x where x>=y>1. OEIS A076980 prepends L(2,1) as its first term. The first ten Leyland number (x,y) pairs ordered by magnitude are (2,1), (2,2), (3,2), (4,2), (...
400 views

### Sum a number's digits until only one digit remain

I want to iteratively calculate the sum of a number's digits, until the result contains only one digit. For example, when $n=67946$ we get: $$6+7+9+4+6=32\space\to\space3+2=5\tag1$$ So, when $n=67946$ ...
112 views

I want to calculate the sum of the product of factorials of the digits of numbers. So for example, when $n=467$ I get: $$n=467\space\to\space\left(4!\right)\cdot\left(6!\right)\cdot\left(7!\right)=... 5 votes 8 answers 1k views ### Making the number 12345...n Well, I am trying to write a code that makes the number:$$123456\dots n\tag1$$So, when n=10 we get:$$12345678910$$And when n=15 we get:$$123456789101112131415$$And when n=4 we get:$$1234...
Well, I have $n\in\mathbb{N}$ and I want to transform $n$ to a binary number which can be done using FromDigits[IntegerDigits[n, 2]]. I want to compute the ...