Articles of integers

sum of square root of primes 2

I dont know how to solve the problem below. (1) $p[1]$, $p[2]$, $\ldots$, $p[n]$ are distinct primes, where $n = 1,2,\ldots$ Let $a[n]$ be the sum of square root of those primes, that is, $a[n] = \sqrt{p[1]}+\ldots+\sqrt{p[n]}$. Show that there exists a polynomial with integer coefficients that has $a[n]$ as a solution. (2) Show that […]

Numbers for which $n!=n^n$

I recently came across this relation valid for all positive integers …which is $n!\leq n^n$ while it’s proof is very easy by basic induction …but I wanted to know for what values of n .. the equality($=$) holds..? I tried putting some values but I found only n=1 is satisfying it…are there any other values […]

Find the five smallest positive integer $W$

Find the five smallest positive integer $W$ of at least two digits with the properties: $W=\frac{(m)(m+1)}{2}$ for some integer $m$. Every digit of $W$ is the same. I was able to find two of the numbers $55,66$ but I did it with a calculator. I’m trying to find a strategy or a pattern but I […]

composition of an integer number

Given two positive integers $m$ and $n$. I would like one special non-negative solution to the following system (which is related to a composition of an integer number): $$\begin{cases} \sum a_i = m \\ \sum (i+1)a_i =n \end{cases}$$ How to describe the solution $(a_0,\dots,a_j)$ with largest index $j$ such that $a_j$ is nonzero? For instance, […]

(Inverse) mapping from $\mathbb{R}$ to subsets of $\mathbb{N}$

There’s the usual mapping $2^\mathbb{N}\mapsto\mathbb{R}_{[0,1)}$ (where $2^\mathbb{N}$ denotes the powerset of $\mathbb{N}$) given by $x\in2^\mathbb{N}\mapsto\sum_{i\in x}\frac{1}{2^i}\in\mathbb{R}$. I’m interested in the inverse $\mathbb{R}_{[0,1)}\mapsto2^\mathbb{N}$, i.e., given $r\in\mathbb{R}$, what’s the corresponding $x_r\in2^\mathbb{N}$ that generates it? (Note: I’m just guessing appropriate tags) Is that $x_r\in2^\mathbb N$ explicitly constructible? Or its characteristic function, i.e., for $r\in\mathbb{R}_{[0,1)}$ and $x_r\mapsto r$, $\chi_r(i)=\left\{{1\ […]

How to solve Diophantine equations of the form $Axy + Bx + Cy + D = N$?

Is there a general solution to solve a Diophantine equation of the form $Axy + Bx + Cy + D = N$? With $A,B,C,D,N,x,y$ positive integers.

Triangle Quadrilateral and pentagon whose areas form a set of consecutive positive integers.

Find a triangle, quadrilateral and pentagon with integer side lengths whose areas form a set of three consecutive positive integers. Make the areas as small as possible subject to these constraints. Report you answer by giving the areas of each figure with the side lengths of those figure? We use a rectangle for our quadrilateral […]

Convert a piecewise linear non-convex function into a linear optimisation problem.

Update: Problem and solution found here (p. 17, 61), although my prof’s solution (formulation) is different. Convert $$\min z = f(x)$$ where $$f(x) = \left\{\begin{matrix} 1-x, & 0 \le x < 1\\ x-1, & 1 \le x < 2\\ \frac{x}{2}, & 2 \le x \le 3 \end{matrix}\right.$$ s.t. $$x \ge 0$$ into a linear integer […]

What is a natural number?

According to page 25 of the book A First Course in Real Analysis, an inductive set is a set of real numbers such that $0$ is in the set and for every real number $x$ in the set, $x + 1$ is also in the set and a natural number is a real number that […]

Find all $a,b,c\in\mathbb{Z}_{\neq0}$ with $\frac ab+\frac bc=\frac ca$

As the title implies, I’m looking for triples $(a,b,c)$, where $a,b,c$ are nonzero integers, with $$\frac ab+\frac bc=\frac ca$$ I checked the cases $-100<a,b,c<100$ where $a,b,c\neq 0$ (using Wolfram Mathematica), and found no solutions. Because of this, I conjecture there’d be no solutions. One idea was using AM-GM, but we barely limit out choice of […]