Articles of number theory

Probability of two events

I am having trouble of calculating the following probability: Let $\epsilon_i$, $i=1,\dotsc,N$ be Rademacher random variables. Let $n_i\in \{0, 1, 2, \dotsc, M\}$, $i=1,\dotsc,N$ such that $\sum_{i=1}^Nn_i=M$. I want to calculate $$ P\left(\left\{\prod_{i=1}^N\epsilon^{n_i}_i=1\right\}\bigcap\left\{\sum_{i=1}^N\epsilon_i=0\right\}\right). $$ Thank you.

Prove that if $a\equiv b \pmod m $ , then $a \bmod m = b \bmod m$

Question Prove that if $a\equiv b \pmod m$ , then $a \bmod m = b \bmod m$ Approach Given, $a\equiv b \pmod m$ $\implies m\mid (a-b)$ $\implies (a-b)=m\cdot k$ $\implies a=b+m\cdot k$ Now, $a \bmod m$ can be written as $\implies b+m\cdot k \pmod m$ No idea how to move forward to get $$b \pmod […]

Finding a (small) prime great enough that there are at least m elements of order m

I’m hoping that someone can provide me with some results or point me in the right direction. I’m working with finite fields; really, I’m just doing arithmetic modulo a prime $p$. I’m taking elements to powers, so I believe this deals with the multiplicative group in particular. Now I basically require that there are at […]

How to reformulate a multiplicative formula (with two primes, perhaps like totient-function)?

In the work on another question in MSE I have a formula $f(n)$ whose pattern depending on $n \in \Bbb N$ I want decode into an algebraical formula (see a short rationale of $f(n) at the end). Beginning with the most simple $n$ – namely that which consist of pairs of primes $p,q$ with $n=pq$ […]

Weak $k$-compositions with each part less than $j$

I am trying to figure out a problem from Richard Stanley’s $\textit{Enumerative Combinatorics}$, which has to do with weak compositions of $n$ (sequence of nonnegative integers whose sum adds up to $n$). The problem is as follows: Let $\kappa(n,j,k)$ be the number of weak compositions of $n$ into $k$ parts, each part less than $j$. […]

Primes and arithmetic progressions

In a book on complex analysis, the authors prove: Given finitely many (non-trivial) arithmetic progressions of natural numbers $$a_1, a_1+d_1, a_1+2d_1, \cdots $$ $$a_2, a_2+d_2, a_2+2d_2, \cdots, $$ $$a_k, a_k+d_k, a_k+2d_k, \cdots, $$ their totality (union) is never $\mathbb{N}$. Given: any $k$ (non-trivial) arithmetic progressions. Let $S$ denote the set of those numbers which are […]

Legendre's Proof (continued fractions) from Hardy's Book

I’m tryng to understand the proof given in Hardys book A Theory of Numbers from the chapter on continued fractions. It states that If $$\left|\frac{p}{q}-x\right| < \frac{1}{2q^2}$$ then $\frac{p}{q}$ is a convegent to $x$. Proof: I the above inequality holds then $$\frac{p}{q} – x = \frac{\epsilon \alpha}{q^2},$$ $\epsilon = \pm 1$ and $ 0 < […]

Simultaneous Diophantine approximation: multiple solutions required

Suppose we have $n$ linearly independent (over $\mathbb{Q}$) irrational numbers $\{ \alpha_i | 1\leq i \leq n \}$. For the simultaneous Diophantine approximation problem $$ |q \alpha_i – p_i | < \epsilon , $$ where $q$ and the $p$’s are all integers, we have the LLL algorithm. The problem is, by this algorithm, for each […]

Solving the Diophantine Equation $2 \cdot 5^n = 3^{2m} + 1$ over $\mathbb{Z}^+$.

Prove that $(m, n) = (1, 1)$ is the only solution for the Diophantine Equation $$2 \cdot 5^n = 3^{2m} + 1$$ where $(m, n) \in (\mathbb{Z}^+)^2$. I’ve managed to prove that both $m$ and $n$ are odd seeing $\bmod 3\text{ and } 10$ respectively. Also, $\forall n \ge 1$, $10$ divides the LHS. I […]

Show $\vert G \vert = \vert HK \vert$ given that $H \trianglelefteq G$, $G$ finite and $K \leq G$.

My problem is a variation of one in Dummit and Foote: Let $G$ be a group and $H \trianglelefteq G$. Prove: If $G$ is finite and $[G:H] = p$, a prime number, then for any $K \leq G$, either $K \leq H$ or $G = HK$ and $[K : H \cap K] = p$. Ok, […]