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 […]

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 $W$ 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 […]

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, […]

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\ […]

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.

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 […]

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 […]

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 […]

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 […]

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