Questions tagged [number-theory]
Questions on the number-theoretic functionality of Mathematica.
441
questions
2
votes
6
answers
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 ...
- 525
5
votes
2
answers
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 ...
0
votes
0
answers
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)
...
- 153
3
votes
1
answer
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 ...
- 1,653
8
votes
4
answers
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 ...
- 1,653
3
votes
4
answers
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 ...
- 145
2
votes
0
answers
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 ...
- 121
1
vote
1
answer
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 ...
- 737
0
votes
1
answer
60
views
Does applying Reduce result in an equivalence or a oneway implication or both? [closed]
If I type something like this into Mathematica:
...
- 1,149
1
vote
1
answer
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 ...
- 245
1
vote
1
answer
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 ...
- 4,037
2
votes
3
answers
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 ...
- 3,188
5
votes
2
answers
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 ...
- 967
3
votes
6
answers
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}
...
- 115
1
vote
1
answer
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 ...
- 93
2
votes
2
answers
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 ...
- 145
5
votes
0
answers
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 ...
- 171
5
votes
4
answers
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:
...
- 6,587
4
votes
3
answers
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 ...
- 5,562
3
votes
4
answers
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 ...
- 53
4
votes
2
answers
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 ...
- 1,941
3
votes
2
answers
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 ...
6
votes
2
answers
109
views
Reduce an expression where the variables can assume only $\pm 1$
I would like to solve this equation:
x1y1 + x2y2 + x3y3 + x4y4 = 0
and I would like to count the number of distict solutions.
Here $x_1,\dots,x_4$ and $y_1,\dots,...
- 239
1
vote
1
answer
96
views
Extracting powers of a sum
Given a sum in the form
Sum[n^k, {k, kl}]
for some natural n and natural list kl, is it ...
- 4,187
0
votes
0
answers
63
views
Repeated DirichletConvolve of MangoldtLambda and MoebiusMu
There is a bug in DirichletConvolve.
These two codes are same except that in first one MoebiusMu is used in another ...
- 7,094
3
votes
0
answers
140
views
DirichletTransform gives incorrect result
Bug introduced in 13.0 or earlier and persisting through 13.2.0 or later.
Input 1:
...
- 7,094
2
votes
0
answers
81
views
Thue equations solving with Reduce
Could somebody tell me whether Reduce assumes (or not) the GRH when solving Thue equations?
Given its performances (related to timing) compared to PARI/GP when the "GRH assumed" flag is set ...
3
votes
2
answers
225
views
Strange behavior of 'SquareFreeQ'
What is the cause of this strange behavior of SquareFreeQ while PrimePowerQ works correctly:
...
- 7,094
2
votes
1
answer
67
views
Choosing numbers whose divisors can be partitioned into subsets having the equal sum
How can we find the list of natural numbers between 100 to 10000, whose positive divisors can be partitioned into three subsets such that sum of all the elements in three subsets will be equal?
- 37
5
votes
1
answer
63
views
Picking integer compositions with certain descent patterns
I am trying to find a nice way to pick out all the integer compositions (ordered partitions) of an integer $n$ that satisfy a given pattern of descents between some adjacent elements. Writing a ...
- 227
4
votes
1
answer
199
views
Choosing a subset of a set based on the sum of its elements
How can we choose a subset of a set based on the sum of the elements of the subset?
For instance,
n=6
dn=Divisors[n]
sn=DivisorSum[n,#&]
Is it possible to ...
- 41
5
votes
1
answer
90
views
How to prove that a list of algebraic numbers with the same symbolic expression is a number field?
Definition of number field (K):
(1) $0,1 \in K$
(2) $\alpha, \beta \in K \Longrightarrow \alpha \pm \beta \ and\ \alpha \beta \in K$
(3) $\alpha, \beta \in K$, and $\beta \neq 0 \Longrightarrow \...
- 1,921
3
votes
2
answers
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):
...
- 1,941
7
votes
2
answers
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 ...
- 245
6
votes
3
answers
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:
...
- 11.6k
2
votes
3
answers
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 ...
- 29
0
votes
1
answer
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?
3
votes
1
answer
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[...
- 737
3
votes
1
answer
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 ...
- 143
0
votes
1
answer
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 ...
- 1,941
2
votes
2
answers
116
views
Finding IntegerPartitions[252] with no zero and no duplicates
Well, I am trying to execute the following code:
...
- 1,941
0
votes
0
answers
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?...
- 1,941
0
votes
0
answers
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 ...
- 101
4
votes
3
answers
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 ...
- 1,941
2
votes
1
answer
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), (...
- 199
5
votes
1
answer
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 $n=67946$ ...
- 1,941
2
votes
0
answers
112
views
Finding the sum of the product of factorials of the digits of numbers
I want to calculate the sum of the product of factorials of the digits of numbers. So for example, when $n=467$ I get:
$$n=467\space\to\space\left(4!\right)\cdot\left(6!\right)\cdot\left(7!\right)=...
- 1,941
5
votes
8
answers
1k
views
Making the number 12345...n
Well, I am trying to write a code that makes the number:
$$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$$...
- 1,941
0
votes
1
answer
85
views
Parallelization in While[] loop when testing if a condition is met
Well, I am trying to run the following code:
...
- 1,941
1
vote
0
answers
100
views
Implementing the Binary GCD algorithm in Mathematica
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 ...
- 1,941