Questions tagged [number-theory]

Questions on the number-theoretic functionality of Mathematica.

454 questions
Filter by
Sorted by
Tagged with
48 views

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 ...
1k views

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 ...
173 views

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: ...
• 4,589
134 views

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 ...
• 155
253 views

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 $a=\frac 54$. ...
• 1,001
496 views

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! ...
• 1,894
75 views

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 ...
• 20.2k
99 views

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 ...
• 381
152 views

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’ ...
• 3
1 vote
118 views

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 ...
3k views

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): ...
• 93
178 views

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, ...
• 923
635 views

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 ...
484 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 ...
• 756
209 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 ...
53 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
165 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 ...
• 1,725
746 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 ...
• 1,725
1k views

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 ...
• 251
47 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
97 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 ...
• 777
60 views

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

If I type something like this into Mathematica: ...
• 1,159
1 vote
88 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
67 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 ...
• 4,325
323 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 ...
• 3,268
645 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 ...
• 1,009
684 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} ...
• 195
1 vote
147 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 ...
187 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
85 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: ...
• 6,747
345 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 ...
• 5,672
107 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 ...
99 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 ...
• 2,021
196 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 ...
116 views

• 2,183
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): ...
• 2,021
287 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
2k 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: ...
• 11.8k
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 ...