# Tagged Questions

58 views

### Number theoretical Application of the Pigeonhole Principle

I'm currently working through a paper related to my bachelors thesis and I'm stuck at a point where the author mentions the following result as "a standard application of the pigeonhole principle". ...
27 views

### Regarding Thue's congruence theorem.

Did the mathematician Thue have a theorem where if $X\cdot N$ is congruent to $y \pmod m$, gcd$(y'm)=1$ then $1 \lt X \le \sqrt{m}$, or $1 \lt Y \le \sqrt{m}$? I'm not sure if I saw this in a number ...
98 views

### Pigeonhole Principle / Number Theory

Let $S$ be a subset of $A=\{1,2,3,...,1000\}$. Find the largest number of elements in $S$ such that for any $a, b \in S$ with $a>b$, $a-b$ does not divide $a+b$. I've tried numerous approaches, ...
119 views

### Pigeonhole Principle - numbers between $1$ and $100$

Of the set $A=${$1,2,...,100$}, we will choose $51$ numbers. Prove that, among the $51$ chosen numbers, there are two such that one is multiple of the other My notes: 1) There are $25$ prime numbers ...
I don't really know how to start this problem at all. I would like a solution or even hints. "Prove that for every odd integer $n$ there is an integer $i$ such that $n \mid 2^i- 1.$" The chapter in ...