Questions tagged [number-theory]
Questions on the number-theoretic functionality of Mathematica.
465 questions
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$LLL$ implementation complexity in wolfram mathematica
$LLL$ is implemented in Wolfram Mathematica as $\mathsf{LatticeReduce}$ command. If we want to reduce a rank $k\leq n$ lattice in $\mathbb Z^n$ where the generator matrix of the lattice has integral ...
- 157
0
votes
1
answer
613
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Find positive-integer tuple satisfying constraints [closed]
I am reading an interesting paper One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive-...
- 373
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1
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DirichletConvolution with an If function returns wrong result
I want to compute a Dirichlet convolution with a function that has a conditional. For simplicity, let's say:
idifeven[n_]:=If[EvenQ[n],n,0]
Let's convolve this ...
- 21
8
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1
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Mathematica code involving floor function
I am reading an interesting paper One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational by Zudilin. Define $$\varphi_0(x,y):=\sum_{j=1}^{3}([y]+[\eta_0x-y]-[y-\eta_j x]-[(\eta_0-\eta_j)x-y]-2[\...
- 373
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How do you increase the precision and accuracy of the numerical approximation of the Volchkov integral?
There is a need to increase the number of correct decimal digits from this integral:
...
- 1,221
10
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1
answer
367
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Recalcitrant Provable Primes
I'm working on an OEIS submission that counts the number of n-digit A157711 primes. Working with PrimeQ as my primality determinator, I can generate some 1200 terms. However, it has been pointed out ...
- 399
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Complete set of 31 numbers which are not the sum of distinct squares
A001422 gives the whole set of 31 numbers which are not the sum of distinct squares (see also MathWorld). That article also provides code for it:
...
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3
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2
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Count the longest streak output
Given a number, I am trying to find the longest streak of positive divisibility into sequential smaller numbers. Here's an example:
We have this 2D list:
...
- 1,747
10
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2
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402
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Having problems with group of units generators
Sorry if the question is quite basic, I started learning Mathematica today to help me with group computations.
I'm trying to get the generators of the group of units modulo p, with p a prime. However, ...
- 103
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Simplifying Computation Subject to Polynomial Equations
I am trying to use Mathematica to simplify sums of the form
$$\sum_{0 \leq j \leq 5}\sum_{0 \leq k \leq 1}\omega^{nj-k},$$ subject to the polynomial equations $\omega^{6} = 1$ and $\omega^{2} = \omega ...
- 13
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Recognizing Euler products
Is there a standard approach in Mathematica for recognizing Euler products? For example, I have the following product
$$\prod_{p} \frac{1-p^{s}+p^{2s}}{\left(p^s-1\right)^2}.$$
There is a nice closed ...
- 213
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0
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How to generate lattice by integrating over curves on a modular form domain plot?
I was watching this video (2:30 of https://youtu.be/zLEyIT_BCgk?si=ji5NyjR7vcwzaNxi ).
Here in the video he said that to get the lattice , we just integrate over the curve(silver line) and we will ...
- 194
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Taxicab Geometry
According to taxicab geometry, there are 1226 possible paths from A(2,5) to B(7,9), all with a distance of 9 units, |2-7|+|9-5|=9. I wanted to write a code where I could be obtaining the plots of some ...
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2
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Are there other methods to list the coprime arrays within the array? [duplicate]
The list array is:
list = Subsets[Range[2, 8], {2}]
get the list is:
...
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1
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Mathematica code to compute a constant
I am reading an interesting paper One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive ...
- 373
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4
answers
333
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Generation of rational numbers under constraints
Working this question on Mathematics Stack Exchange, I have the following questions:
how to generate a rational number $a$ such that $\sqrt{a^2-1}$ be also rational? This is the case of $a=\frac 54$.
...
- 1,105
4
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5
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588
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How to check if multiplication requires carries?
For some purposes I need to know if there are there any carries in the multiplication of two numbers, especially in base-2.
How can we do this in Mathematica?
Thanks
to all, very interesting answers!
...
- 2,590
3
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0
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96
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PrimeZetaP evaluation in different versions of Mathematica
PrimeZetaP was introduced in version 7.0. I suspect there were made some changes in the definition of this function in subsequent versions.
Is there any user that ...
- 32.8k
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How to find value of variables so that my expression to be perfect squares?
Suppose I have the expression
$$\sqrt{p(2-p)} \tag 1$$
and the expression
$$
\sqrt{\frac{1}{4}\left( p-2 \right) ^2-\frac{4\left( p-1 \right) ^4}{\left( p-2 \right) ^2}}. \tag 2
$$
The Mathematica ...
- 521
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1
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PartitionsP for non-integer arguments [closed]
Is there a reason why the function PartitionsP[] does not return anything for non-integer arguments, although there is a way to calculate using Rademacher’s ‘exact’ ...
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Does a(a+1)(a+2)-1=b²+2=10c+3 have solutions over natural numbers? If yes, how many?
It is a question from the math olympiad I was participating in that happened like a month ago. The provided solution turned out to be wrong.
It isn't the question itself, but the solution basically ...
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Digits of Pi in colored spiral
In How to make the digits of π go around in a spiral like this?
it is described how to plot pi in a spiralform (in my case as binary number):
...
- 103
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2
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Select a subset in the bit-strings with even 1s overlapped - thank you
Model 1
Consider the permutation list of 4-bit-strings:
list = Permutations[{0, 0, 1, 1}, {4}]
which outputs:
{{0, 0, 1, 1}, {0, 1, 0, 1}, {0, 1, 1, 0}, {1, 0, 0, ...
- 963
8
votes
2
answers
712
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Better code for Ramsey partitions
Ramsey partitions for parameters $a$ and $b$ are certain integer partition of $a+b$ developed by McAvaney, Robertson, and Webb (Combinatorica 1992) for applications in fair division problems where two ...
- 319
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6
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659
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All solutions that satisfy $ x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}-x_{5}^{5}=0 $
I want to find a combination that satisfies all the solutions of the following formula.
$$
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 ...
- 776
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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 ...
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97
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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)
...
- 181
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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 ...
- 1,885
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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 ...
- 1,885
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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 ...
- 647
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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 ...
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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 ...
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Does applying Reduce result in an equivalence or a oneway implication or both? [closed]
If I type something like this into Mathematica:
...
- 1,221
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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 ...
- 263
1
vote
1
answer
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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 ...
- 4,435
2
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3
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369
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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 ...
- 3,303
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2
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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 ...
- 1,061
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8
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744
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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}
...
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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
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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
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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 ...
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4
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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:
...
- 6,807
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3
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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 ...
- 5,782
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4
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267
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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
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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 ...
- 2,021
3
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3
answers
248
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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
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2
answers
161
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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
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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 ...
- 5,023
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vote
0
answers
100
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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 ...
- 32.8k
3
votes
0
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177
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DirichletTransform gives incorrect result
Bug introduced in 13.0 or earlier and persisting through 13.2.0 or later.
Input 1:
...
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