# Tagged Questions

Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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### Confusion between Sequences and Number theoretic functions.

I've just started learning Number Theoretic function,the definition of ,Number Theoretic function,which i've just read created some confusion b/w Number Theoretic function & Sequences. The ...
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### Find the $k$ such that $2^{(k-1)n+1}$ does not divide $\frac{(kn)!}{n!}$.

Find all positive integers $k$ such that for any positive integer $n$, $2^{(k-1)n+1}$ does not divide $\frac{(kn)!}{n!}$. From olympiad problem I'm curious So far no one to solve this problem,Maybe ...
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### Is the difference between $2$ coprimes always either 1 or prime number?

If a and b are both positive and coprime with $a > b$, Is $a - b$ always either $1$ or prime number? Can $a - b$ be composite number?
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### How to explain the rationality of a solution?

This an exercise from an elementary number theory textbook: "The curve $$y^2 = x^3 + 8$$ contains the points $(1, -3)$ and $(-7/4, 13/8)$. The line through these two points intersects the curve in ...
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### Even Digit Series (2,4,6,8,20,…) [duplicate]

I have a series of numbers whose Nth term is a number whose all of the digits are even. The series is of course divergent , but i am interested in finding a formula to find the Nth term of this series....
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### Showing that there are infinitely many integer solutions for the hyperbolic formula $|a^2 - 26 b^2| = 1$

I want to show that the formula $$| a^2 - 26\cdot b^2| = 1$$ has infinitely many solutions $(a, b) \in \mathbb{Z}^2$. First I tried to solve the formula for one of the two variables, to get ...
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### Use of greatest common divisor to calculate unknown

We have three numbers $x ,y, z$. If we know the values of $x$ and $z$ then is it correct to say that $y$ should be a multiple of $z/\gcd(z,x)$ for the expression shown below to be true? Here $\gcd$ ...
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### Skiponacci: $p | a_p$ Alternate Solution

For the Skiponacci sequence: $a_0=3, a_1=0, a_2=2,$ and $a_{n+1}=a_{n-1}-a_{n-2}$ for $n>2$, prove that any prime $p$ divides $a_p$. Is there any alternate solution other than using ...
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### Basic question with coprimes and modulos

I started reading about Modular Arithmetic and solving some random basic exercises, and this one appeared: "Find an integer number $a$ such that any $b$ coprime with 34 is congruent to $a^k \mod34$ ...
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### Origin of Almost Perfect Numbers

Let $N$ be a positive integer. $N$ is called a perfect number if the sum of its positive divisors denoted by $\sigma(N)=2N$. For example $6$ is a perfect number since: $\sigma(N)=1+2+3+6=12=2(6)$. ...
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### System of linear congruence when not relatively prime

I am new to Abstract Algebra and understand how to solve when the mods are relatively prime, but I am struggling when they aren't relatively prime. I have a system of of linear congruences that I ...
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### If $d=\gcd\,(f(0),f(1),f(2),\cdots,f(n))$ then $d|f(x)$ for all $x \in \mathbb{Z}$

$\textbf{Question.}$ Let $f$ be a polynomial of degree $n$ which takes only integral values. If $d=\gcd\,\{f(0),f(1),f(2),\cdots,f(n)\}$ then show that $d|f(x)$ for all $x \in \mathbb{Z}$. How ...
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### Analogy to Four Squares Theorem.

Is there a multivariate and univariate polynomial analogy to Lagrange's sum of four squares?
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### Cyclic prime groups

can I have a refrence to an introduction (not super beginner level, one after) of the multiplicative group $Z/ZP$? I know that it is cyclic. I am interested in known properties of the generators. ...
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### Succession in Peano axioms

In "Analysis I" - Herbert Amann states: "The natural numbers consist of a set $N$ , a distinguished element $0\in N$ and a function $v:N\to N^*$ with the following properties: ($N0$) $v$ is ...
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### Finding unknown numbers using $LCM$ and $HCF$

Find two numbers, $A$ and $B$, both smaller than $100$, that have a lowest common multiple of $450$ and a highest common factor of $15$. I know that this involves the formula of $A × B = LCM × HCF$ ...
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### What am I missing in this induction proof?

Prove that if $g:\mathbb{N}\rightarrow \mathbb{N}$ and $\forall x,y\in \mathbb{N}, x<y\Rightarrow g(x)<g(y)$ then $n\leq g(n)\space\space\space \forall n\in \mathbb{N}$ My proof so far (...
### How to find $k$-th number whose digits are all even?
As my question says I have to find $k$-th number whose digits are all even. I figure out that all those numbers are made of of $\{0,2,4,6,8\}$ and there is a sequence in which the numbers change their ...