Articles of number theory

To solve the system of Diophantine equations.

I decided to compile a single task and to record such a system. $$\left\{\begin{aligned}&xt+yw=az^2\\&xw-yt=br^2\end{aligned}\right.$$ $a,b – $ integers that are the problem. It is clear that it is necessary to solve the problem to get the formula. For $x,y,t,w,z,r – $ which is described in integers. And at the same time to find out all […]

Dirichlet hyperbola methods : estimate functions # of ordered pairs

Let $f$ be an arithmetic function defined by $$f(n) = |A_n|$$ where $A_n = \{(a, b) : n = ab^2\}$.Estimate $$\sum_{n \leq x} f(n)$$ where $x \in \mathbb{R}^+$, using Dirichlet hyperbola method. The error term should be $O(x^{1/3})$. Dirichlet hyperbola method requires the function $f$ to be written as Dirichlet convolution of two functions. The […]

Coefficient in the Fourier expansion of the cusp form

Ideal of cusp for $\Gamma_{0}(4)$ is principal and generated by $f(z)=η(2z)^{12}=q+\sum a(n)q^n $, this is discussed here. How one can compute the coefficient $a(n)$ when $n$ is rather large ? for example, what is the coefficient $a(2015)$ ?

Consecutive numbers with less than $k$ prime factors?

Let $k$ be an integer. Consider the consecutive numbers with less than $k$ distinct prime factors. Are there arbitary large differences between those numbers ? With other words : Are there arbitary many consecutive numbers having at least $k$ distinct prime factors for every natural number $k$ ? For $k=3$, the jumping champions are : […]

Prove that the expression has infinitely many solutions

Does $m = \dfrac{1}{72}\left(\sqrt{48r^2+1}+1\right)$ have infinitely many solutions in positive integers $(m,r)$? Does $m = \dfrac{1}{18}\left(\sqrt{48r^2+1}-1\right)$ have infinitely many solutions in positive integers $(m,r)$ where $m \equiv 3 \pmod{4}$? I wasn’t sure how to prove that there are infinitely many solutions, but for both of them we need $48r^2$ to be one less than a […]

Number of divisiors of $n$ less than $m$

I’m looking for a closed- or alternative-form of the function that counts the number of divisors of an integer $n$ that are less than some integer $m$ (interested in $m < n$, obviously): $ \sigma_0(n,m) = \sum_{d|n,d<m} 1 $. Or more generally: $ \sigma_x(n,m) = \sum_{d|n,d<m} d^x $. Or any important property of either (like […]

How prove this irrational $x\in(0,1)$,then have $0<x-\sum_{i=1}^{n}\frac{1}{p_{i}}<\frac{1}{n!(n!+1)}$

show that: for any irrational $x\in(0,1)$,and positive integer $n$,there exsit positive integer $p_{1},p_{2},\cdots,p_{n}$ where $$p_{1}<p_{2}<\cdots<p_{n}$$ such $$0<x-\sum_{i=1}^{n}\dfrac{1}{p_{i}}<\dfrac{1}{n!(n!+1)}$$ This problem is from Mathematical contest in jiangxi province at last problem. I think this reslut maybe involves Irrational Approximation? Note $$\dfrac{1}{n!(n!+1)}=\dfrac{1}{n!}-\dfrac{1}{n!+1}$$ Thank you for you help

Powers of $x$ as members of Galois Field and their representation as remainders

first question on math.stackexchange 🙂 I’m studying for a Cryptography – Communication Security exam, and it involves a certain quantity of number theory – finite field theory, so be warned: this is my first encounter with these topics, and you’ll have to be extra-clear with me 🙂 I thought I was doing pretty well with […]

Another Watered Down Version of Dirichlet's Theorem

Dirichlet’s theorem states that given integers $a,b$ with $(a,b)=1$, there are infinitely many primes $p$ such that $p \equiv a \pmod b$. Much has been made of special cases of this theorem that are easy to prove without analysis, but suppose one only needed to know that there exists such a prime (i.e. replace “infinitely […]

Convergence of the sum of two infinite series only at $x=\frac12$?

I would like to ask a simplified version of this question on MO: question about infinite series Assume $x \in \mathbb{R}$. I like to conjecture that the sum of the following two infinite series: $$\displaystyle Z(x) = \frac{1}{2\,(x-1)} \sum _{n=1}^{\infty } {\frac {x-1-2\,n}{{n}^{x}}} + \frac{1}{2\,(-x)} \sum _{n=0}^{\infty } {\frac {1-x+1+2\,n}{\left( n+1 \right) ^{1-x}}}$$ only converges […]