# 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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### 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 ...
1answer
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### 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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### 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 ...
1answer
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### 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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### 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 ...
1answer
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### 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 ...
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### 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?
1answer
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### 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}} \...
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### 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.
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### 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 ...
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### 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 ...