Articles of number theory

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 […]

Are there infinitely many nonnegative integers not of the following four forms?

The four forms are: $3x^2 + (6y-4)x – y$ $3x^2 + (6y-2)x + y – 1$ $3x^2 + (6y-3)x + y – 1$ $3x^2 + (6y-3)x – y\ $,$\ x,y \in \mathbb{Z}^{+}$ For example: $4=3 \cdot1^2+(6 \cdot1-4) \cdot 1-1\ $ is the minimum number of the four forms, so that $\ 0, \ 1, \ […]

Additive identity in $\mathbb{Z}_{7}$

The problem consists of two parts: 1)Prove that for some integer $m,$ $3^{m} = 1$ in the integers modulo 7, $\mathbb{Z}_{7}$, which i have done. The second part asks to use this fact to deduce that $7$ divides $1 + 3^{2001}.$ The way i look at it, we look at $1 + 3^{2001}$ in $\mathbb{Z}_{7}$ […]

Related to the partition of $n$ where $p(n,2)$ denotes the number of partitions $\geq 2$

I was trying the following question from the Number theory book of Zuckerman. If $p(n,2)$ denotes the number of partition of n with parts $\geq2$, prove that $p(n,2)>p(n-1,2)$ for all $n\geq 8$, that $p(n)=p(n-1)+p(n,2)$ for all $n \geq 1$, and that $p(n+1)+p(n-1)>2p(n)$ for all $n \geq 7$. Need some clue to solve it.

Sum of four squares modulo a prime

With $p$ a prime integer, I am interested in showing the existence of four non-negative integers $\alpha_0,\alpha_1,\alpha_2,\alpha_3$, not all zero, such that $$\alpha_0^2+\alpha_1^2+\alpha_2^2+\alpha_3^2\equiv 0\mod p.$$ I am, of course, aware of the Lagrange four square theorem of which this is a special case. However, the fact that I am concerned with only prime $p$, and […]

Must a certain continued fraction have “small” partial quotients?

I have reformulated the original question, which appears at the bottom, it a way that seems more likely to produce a reference. New version: Let $\Delta$ be a positive nonsquare integer congruent to $0$ or $1$ modulo $4$. If $\Delta$ is even, expand $\frac{\sqrt{\Delta}}{2}$ in a simple continued fraction. If $\Delta$ is odd, expand $\frac{\sqrt{\Delta}+1}{2}$ […]

On the distribution of “nice” primes (primes $p$ , such that $\pi(p)$ is prime as well)

Define a ‘nice prime’ by a prime such that it is the $n$th prime where $n$ is a prime number. For example, $3$ is a nice prime since it is the $2$nd prime and 2 is a prime itself. $5$ is also a nice prime since it is the $3$rd prime and $3$ is prime. […]

Does a random binary sequence almost always have a finite number of prime prefixes?

Does a random binary sequence almost always have a finite number of prime prefixes? Specifically, let $x = \sum_{1 \le i}{2^{-i} \cdot x_i}$ with $x_i \in \{0,1\}$ be a random real in $[0,1)$, $X_i = \sum_{0 \le j \le i}{2^{i-j} \cdot x_j}$, and $F(x) = \cup_{i}{\{X_i\}}$. How can we prove that F(x) almost certainly does […]

How do I find an integral basis, given a basis consisting of algebraic integers?

A known example of a number field that has no power basis is the field $\mathbb{Q}(\theta)$, where $\theta$ is a root of the polynomial $x^3-x^2-2x-8$. The discriminant of this polynomial is $-2012 =-503 . 4$ and also is the discriminant of the basis $(1, \theta, \theta^2)$. Now it appears that the basis $(1, \theta, (\theta+\theta^2)/2)$ […]