Articles of elementary number theory

How to prove divisibility test for $4$?

This question already has an answer here: Divisibility test for $4$ 3 answers

There are infinitely many triangular numbers that are the sum of two other such numbers

In the Exercise $9$, page 16, from Burton’s book Elementary Number Theory he state the following: Establish the identity $t_{x}=t_{y}+t_{z},$ ($t_{n}$ is the nth triangular numbers) where $${x}=\frac{n(n+3)}{2}+1\,\,\,\,\,\,\,y=n+1\,\,\,\,\,\,\,z=\frac{n(n+3)}{2}$$ and $n\geq 1,$ thereby proving that there are infinitely many triangular numbers that are the sum of two other such numbers. I tried to find out how […]

An intuitive interpretation of Montgomery pair corrlation function vs. prime divisibility?

Theorem – If the Riemann hypothesis would be true, and the Montgomery pair correlation conjecture (see linked article page 183-184) true too; let $p \in \Bbb P$ prime, $n \in \Bbb N$ and $$\mathcal V_p(n)=1-(n^{p-1}\mod p)$$ then$$\lim_{p\rightarrow \infty}\mathcal V_p(n) = \operatorname{sinc}(2\pi \,n)$$ and for the pair correlation of the non-trivial zeroes of the Riemann $\zeta$-function, […]

Prove that if $n^2$ is even then $n$ is even

Assume that $n^2$ is even Therefore $n^2 = 2k$ for some integer $k$. How do I finish this proof?

finding units of $ \mathbb{Z} {3}] $

In order took for units of $ \mathbb{Z} [ \sqrt[3]{3}] $ I am using a generalized Euclidean algorithm on three numbers. If $x \leq y \leq z$ then : $$ (x,y,z) \to \text{ sort } ( x, y ,z -y ) $$ The three numbers I will use are $1, \sqrt[3]{3},\sqrt[3]{9}$. Then if $z-y$ is […]

prove that there are infinitely many numbers of the form $x = 111…1$ such that $31|x$

I need to prove that there are infinitely many numbers of the form $x = 111….1$ such that $31|x$ what i tried – I wrote x as $\sum_0^{n-1} 10^i$ i know that $(10,31) = 1 $ now im stuck .. any help will be aprriciated

Arithmetical proof of $\cfrac{1}{a+b}\binom{a+b}{a}$ is an integer when $(a,b)=1$

When $(a,b)=1$, $\cfrac{1}{a+b}\binom{a+b}{a}$ refers to the number of paths from one corner to its opposite corner of an $a\times b$ lattice that lies completely above (or below) the diagonal. Therefore, it must be an integer. But does anyone know if there is an arithmetical proof of this? There is an arithmetical proof for $\binom{a}{b}$ is […]

Distribution and expected value of a random infinite series $\sum_{n \geq 1} \frac{1}{(\text{lcm} (n,r_n))^2}$

Can we find the distribution and/or expected value of $$S=\sum_{n \geq 1} \frac{1}{(\text{lcm} (n,r_n))^2}$$ where $r_n$ is a uniformly distributed random integer, $r_n \in [1,n]$ and $\text{lcm}$ is the least common multiple? Or maybe an estimate is possible? Some boundaries are easy to see. For example: $$1<S \leq \sum_{n \geq 1} \frac{1}{n^2}= \frac{\pi^2}{6} \approx 1.645$$ […]

Looking for a very gentle first book on number theory

My gf is the classic math-phobe, totally traumatized by math, etc., which surprises me, since she’s whip-smart. The only explanation I can think of is that she got off to a bad start in elementary school. My own love for mathematics was kindled by an old book on number theory that I found in my […]

gcd as positive linear combination

Good evening, I have a question concerning the euclidean algorithm. One knows that for $a_1 , \ldots , a_n \in \mathbb{N} $ and $k\in \mathbb{N} $ there exist some $\lambda_i \in \mathbb{Z}$ such that : $$\gcd(a_1, \ldots, a_n) = \frac{1}{k}\sum_{i=1}^n \lambda_i a_i$$ Here is my question: can one find a $m_0 \in \mathbb{N}$ that for […]