Questions tagged [number-theory]

Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebra geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.

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Number Theory: Solving Linear Congruence

Come up with a counterexample to Proposition 5.2.7. when $\gcd(d,n)≠1$. (Proposition 5.2.7. Canceling, Part II: If $d≠0$ and $\gcd(d,n)=1$,then $ad≡bd \pmod n$ precisely for the same $a,b,n$ as ...
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2answers
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how do I calculate the Thue-Morse-Sequence over the alphabet {0,1} for $\left(w_{2021}\right)_{2} \bmod 19$?

We define the Thue-Morse-Sequence over the alphabet $\Sigma:=\{0,1\}$ as follows: we set $w_{0}:=0$, and for $n \in \mathbb{N}$ we define $w_{n+1}:=w_{n} \overline{w_{n}}$, where $\bar{w}$ is the unit ...
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1answer
18 views

How to receive decimal representation from Cauchy-sequence definition of $\mathbb{R}$

During completing my Bachelor's degree, I sometimes put very few effort into understanding very fundamental, almost philosophical questions. This is because to me it always sufficed to think of a real ...
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1answer
44 views

Quaternion with only two imaginary numbers

Why do quaternions not only use two imaginary numbers. Can we not simplify quaternions $$q = a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k} \tag{1}$$ to the form $$ \begin{align} q & = a + b\mathbf{i}...
2
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0answers
52 views

Is $\sum_{p\text{ is a prime}}\frac{1}{p^{2}}$ known to be irrational/transcendental? [duplicate]

Is $\sum_{p\text{ is a prime}}\frac{1}{p^{2}}$ known to be irrational/transcendental? The sum exists, because $\sum_{p\text{ is a prime}}\frac{1}{p^{2}}\leq\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{\pi^...
4
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0answers
71 views

Is this “classic probability view” on Fermat Last Theorem studied somewhere? (re-ask of a deleted question)

This is a re-ask of a deleted question that OP doesn't really wanna undelete but has given me permission to re-ask: https://math.stackexchange.com/questions/2863781/is-this-classic-probability-view-on-...
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1answer
17 views

Finding integer solutions of K for equation floor(A/K) = B

How to find integer solutions of K for equation floor(A/K) = B, in terms of A and B where A and B are non-negative integers? What I tried: floor(A/K) = B then B <= A/K < B + 1 then BK <= A &...
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0answers
25 views

Ramification question in compositum of cyclotomic and degree 5 extension.

I am reviewing some past exam questions and I have a problem solving the following example: Let $F = \mathbb{Q}(\zeta_5, \sqrt[5]{75})$, a field of degree 20 over $\mathbb{Q}$. Determine the ...
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1answer
75 views

Does this Set Cover $(0,1] \cap \mathbb Q$?

Let $\{\frac{1}{n}\}_{n \in \mathbb N}$ be a sequence in $\mathbb Q$. Now in each interval $(\frac{1}{n+1},\frac{1}{n}]$ define a similar sequence $$ \left\{\frac{1}{n+1}+\frac{\frac{1}{n}-\frac{1}{n+...
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2answers
44 views

Generalized Understanding of “Switch” Problem

Lets assume you have a room of 10 switches that are all turned off. You also have 10 robots named 1-10. Starting with robot #1, it goes over and flips all of the switches divisible by its name. When I ...
3
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1answer
100 views

Find all pair of primes $(p,q)$ such that both $p^2+q^3$ and $p^3+q^2$ are perfect squares.

Let $p^2+q^3=a^2$ and $p^3+q^2=b^2$. Let's suppose $ p \neq q$. When one of $p,q$ equals $2$, it yields system of equations with no solution, so $p,q \geq 3$. Since any two primes numbers are coprime,...
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24 views

least upper bounds that are coprime

Given $n$ natural numbers $p_1$, $p_2$, ... $p_n$ find numbers $q_1$, $q_2$, ... $q_n$ that are pairwise coprimes such that $p_i$ ≤ $q_i$ and such that $\prod_{i=1..n} q_i$ is smallest possible. I ...
3
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1answer
50 views

Combinatorial Necklaces & Strips of $n$ Beads and $k$ Colours

Say I have $n$ indistinguishable beads and $k$ different colours. Suppose here and for the rest of the writeup that $k \mid n$ unless otherwise stated. I want to colour all the $n$ beads using exactly ...
5
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1answer
69 views

Kummer theory as an equivalence of categories

The main theorem of Galois theory (in the finite case, for convenience) says that Theorem 1: Let $L/k$ be a finite Galois extension with Galois group $G$. The maps $$M\mapsto \operatorname{Aut}(L|M)\...
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27 views

Modular function for upper triangular matrices from Iwaniec

In Section 1.2 of Iwaniec's Spectral Methods of Automorphic Forms, there is a description of Haar measures and modular functions. There is a point where he deduces the modular function for the group ...
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1answer
41 views

Proof of THE BROCARD PROBLEM. [closed]

I found the following proof in internet ... please someone specialist in number theory give his feedback about this proof... https://drive.google.com/file/d/1lczuhxKXJ7zrxDFlQ4p2TcIaM2azGxo4/view?usp=...
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0answers
22 views

Given $n\in\mathbb N$ what are the maximum number of divisors of a natural number not more than $n$? [duplicate]

So the question is in the title itself. The minimum number of divisors are $2$ (for primes), but the maximum number of divisors are what we are interested in. It cannot be more than the number itself ...
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0answers
24 views

Which sets of $n-1$ non-multiples of $n$ can't make a multiple of $n$ using $+,-$?

This is a follow up to my previous question (see linked question). In short, there it is shown that if $n$ is prime, then any set can make it. I want to characterize sets $\mathbb A_n$ of multisets $...
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1answer
38 views

Find a minimal lcm

Suppose $N$ positive integers, $x_1,x_2,\cdots,x_N$, satisfy $x_1+x_2+\cdots+x_N=K$, where $K$ is a positive contant. Now, I want to know how to find the minimal $L\triangleq \text{lcm}\left(x_1,x_2,\...
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1answer
23 views

Why do triangles in multiple dimensions use pascals triangle

If we take a triangle in 0 dimensions, and we count the number of points it has, we see it only has 1 (since anything in 0 dimensions is a point) |Points| |:----:| |1| If we take a triangle in 1 ...
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0answers
29 views

Proof of these two lemma in analytical number theory

in publication On Gaps Between Primitive Roots in the Hamming Metric by Dietmann, Sharplinski & Elsholtz, there are 2 lemmas: H is a binary entropy function. Unfortunately these lemmas are not ...
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13 views

Convergence in different numeral systems depends on prime factors

I'm sorry that the title isn't good enough but I'm not sure what to write there. I found the following property: Let $1 < k, l \in \mathbb{N}$. Let $\overline{x}_k = (x_0, x_1, x_2, \ldots)_k$ be ...
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0answers
16 views

Is it true that any fraction a/b, gcd(a, b) = 1, is a terminating decimal iff b at most has 2 and 5 in its prime factorization? [duplicate]

Is it possible to prove? I believe the answer is yes; that iff b is of the form 2^c * 5^d for non-negative integers c and d, then a/b has a terminating decimal expansion.
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1answer
20 views

Sets $S$ such that no difference of two distinct elements is divisble by a square

In this paper, Ruzsa shows that for $n$ sufficiently large, there exists a set $S$ of size $\Omega (n^{0.733})$ such that for all $a,b\in S$, the number $a-b$ is not a square. I am wondering about a ...
2
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1answer
53 views

On Erdős' elementary proof of the asymptotics of the partition function

My question concerns a step in Erdős' "elementary" proof of $p(n) \sim \frac{a}{n}e^{cn^{\frac{1}{2}}}$, where $p(n)$ is the number of partitions of a natural number $n$ and $c=\pi\sqrt{(2/3)...
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0answers
27 views

Please help prove this relation.

Let a be an integer such that $1\le a\le n$ and gcd(a, n) = 1. Prove that the product of all the integers a can be given by the relation $n^{\phi(n)} \prod_{d|n} (\...
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0answers
19 views

Relation between discriminant matrices

Suppose that $K = \mathbb{Q}(\xi_1)$ is a number field of degree $n$ over $\mathbb{Q}$ for some $\xi_1$ in the ring of integers of $K$. Suppose that $\{\sigma_1,\ldots,\sigma_n\}$ are the distinct ...
6
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0answers
76 views

What is the density of numbers which have at least two divisors whose sum is a perfect square?

A positive integer is said to have square-sum divisors if it has at least two divisors whose sum is a perfect square. $6$ has square-sum divisors because its divisors are $(1,2,3,6)$ and $1 + 3 = 2^2$...
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33 views

The Coordinate Equation & Fermat's Last Theorem

The Coordinate Equation & Fermat's Last Theorem http://www.urmt.org/Coordinate_Eqn_FLT.pdf Abstract This paper presents a Diophantine equation, known as the Coordinate Equation, which is identical ...
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2answers
28 views

If $\frac{7}{2^{1/2} + 2^{1/4} + 1} = A + B*2^{1/4} + C*2^{1/2} + D*2^{3/4},$ find $A,B,C,D$

If $\frac{7}{2^{1/2} + 2^{1/4} + 1} = A + B*2^{1/4} + C*2^{1/2} + D*2^{3/4},$ find $A,B,C,D$ . What I Tried: I wrote :- $$\rightarrow \frac{7}{2^{1/2} + 2^{1/4} + 1} = \frac{2^2 + 2 + 1}{2^{1/2} + 2^{...
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0answers
20 views

Fermat Pseudoprime when (b, n) is not 1 [closed]

If $b^{n-1} \equiv 1 \pmod n$, then $\gcd(b, n) = 1$ (where $b > 1$ and $n$ is odd composite) is it true?
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1answer
31 views

Asymptotic equivalence of sum of arithmetic function

Let $f$ be an arithmetic function such that $\sum_{n \leq N} f(n) n^{-\frac{1}{2}} \sim N$ as $N \rightarrow \infty$. Prove that \begin{equation} \sum_{n \leq N} f(n) \sim \frac{2}{3} N^{\frac{3}{2}} \...
3
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1answer
82 views

find all triplets $(a,b,c)$ such that $ab+bc+ca=1$ and $a^2b+c=b^2c+a=c^2a+b$.

Given that $a,b,c \in R$ find all triplets $(a,b,c)$ such that $ab+bc+ca=1$ and $a^2b+c=b^2c+a=c^2a+b$. Attempt: Case $I:$ Exactly one of $a,b,c$ is zero. WLOG, let $a=0$. Then we have $$bc=1$$ and $$...
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1answer
14 views

Estimate the sum of expansions for fractional part function

Are there any estimates for the sum of Fourier series expansions for fractional part function?: \begin{equation} \sum_{x=1}^{n}\sum_{k=1}^{\infty}\frac{\sin(2\pi k \frac{n}{x})}{k}, \; n \in \mathbb{...
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0answers
72 views
+50

Find the $d$ such $(d^2+d)x^2-y^2=d^2-1$ has postive integer $(x,y)$

find the postive integer $d$ such $$(d^2+d)x^2-y^2=d^2-1$$ have postive integer solution $(x,y)$ maybe use Pell equation some result $$x^2-Dy^2=C$$ to solve it?
1
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3answers
68 views

Show that the equation $\frac{1}{2} =4x^3-3x$ has no rational root.

Suppose it has rational root, then $x=\frac{p}{q}$, where $q\neq 0$ and $(p, q)=1$. Then the equation can be written as, $$q^3=2(4p^3-3pq^2)$$ So $q$ is even, this force $p$ to be an odd. So we can ...
1
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2answers
81 views

Proof of $ d(n)\leq \sqrt {3n}$.

Show that $$ d(n)\leq \sqrt {3n}$$ and the equality is true if only if $n=12$, where $d(n)$ is the number of positive divisors of $n$.. Here is a proof Proof of $ d(n)\leq \sqrt {3n}$. Let $$ n=\...
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1answer
71 views

Suppose that today is Monday. What is the day of week after $2021^{4019^{419}}$ days?

I know that I have to use $2021^{4019^{419}}\pmod7$ to solve this question, but I don't know to how to further develop.
2
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0answers
41 views

Given $n\in\mathbb{N}_{\geqslant 2}$, find the partition $(a_1,…,a_k)\in\mathbb{N}^k:\sum_{i=1}^k a_i=n$ of $n$ that maximizes $\prod_{i=1}^k a_i$

I am a solving programming question in Leetcode in which, given a number $n \in \mathbb{N}_{\geqslant 2}$, I have to find $(a_1, ..., a_k) \in \mathbb{N}^k$ such that $k \in \mathbb{N}$, $2 \leqslant ...
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0answers
39 views

Finding Lattice Points on a Sphere

I want to find all the lattice points on a sphere with integer radius, or equivalently all solutions to the diophantine equation $x^2+y^2+z^2=r^2$. We can see from oeis that the number of solutions, ...
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1answer
48 views

Euler phi function proof [duplicate]

Let $a,e$ and $f \in \Bbb{Z}^+$ and p is prime number. If $\phi(p^e) | f$ then show $a^f \equiv 1 \pmod{p^e}$ $(gcd(a,p^e)=1$
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0answers
62 views

Minimum of a function connected to permutations weighing by inverse squares

The question (may be of classical number theory and combinatorics) arose during my study, please help me Let $ K,T$ be two natural numbers such that, $1\leq T\leq K!$. Consider the set, $S=\{f:\{1,2,.....
10
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2answers
250 views

Integers less than $7000$ achievable by starting with $x=0$ and applying $x\to\lceil x^2/2\rceil$, $x\to\lfloor x/3\rfloor$, $x\to9x+2$

Robert is playing a game with numbers. If he has the number $x$, then in the next move, he can do one of the following: $\bullet$ Replace $x$ by $\lceil{\frac{x^2}{2}}\rceil$ $\bullet$ Replace $x$ by $...
4
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1answer
136 views

Choosing $16$ integers from first $150$ integers such that there no $(a,b,c,d)$ for which $a+b=c+d$

In how many ways one can choose $16$ distinct positive integers from first $150$ positive integers such that there are no $4$ distinct ones $(a,b,c,d)$ for which $a+b=c+d$? I'm confused with the ...
1
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0answers
29 views

How to represent smallest prime factor of a number in terms of known functions?

The question is in the title. Given a number $n\in\mathbb N\setminus\{1\}$, I wish to write a function $\alpha(n)$ which denotes the smallest prime factor of $n$, in terms of functions which have been ...
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2answers
39 views

Can you explain this simply: $(a + b + c + \dotsc)^p = a^p + b^p + c^p + \dotsc + M(p)$?

I also have a related question to this. I found out that $(a+b+c+\dotsc+n)^2 = a^2 + b^2 + c^2 + \dotsc + n^2 + 2$ (each number multiple once with itself but not considered again). E.g. $(a+b+c+d)^2 = ...
0
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1answer
60 views

Improving Simple Probabilistic Prime Model To Model Prime Gaps

I am developing an intentionally simple probabilistic model of primes with the aim of modelling experimentally measured prime gaps. Question: My simple model has the right shape, and broadly has the ...
0
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1answer
44 views

Determine the number which is crossed out twice in a polygon of 2000 vertices.

Suppose we have a regular polygon with 2000 vertices which are labeled as the natural numbers 1, 2, 3, ... etc. First, we cross out 1. Followed by, the next numbers which are crossed out are $16, 31, ...
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1answer
44 views

proving that an equation has infinitely many solutions

I have the equation $(4n+3)^2-48m^2=1$ that I changed into $X^2-48m^2=1$ so that it would be a Pell's equation, that has infinitely many solutions, that I found being $\left\{\begin{align} x_k = \...
1
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1answer
50 views

Class field theory for $\mathbb{Q}_p$

Let $K$ be a local field and fix an algebraic closure $\bar{K}$. Local class field theory says basically that $L\mapsto N_{L/K}(L^\times)$ is a order-reversing bijection between the finite abelian ...

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