Articles of integers

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 […]

Can the integers be made into a vector space over any Finite Field?

Given a Finite Field $F$, can the the abelian group $\mathbb Z$ be made into a vector space over $F$ without changing the additive structure of $\mathbb Z$? This seems like it shouldn’t be complicated, but having trouble on this one, any hints would be appreciated. I already know that $F\cong \mathbb F_{p^n}$ for some […]

Find all possible values of $x+y+z$.

Let $x, y, z$ be non-zero integers such that ${x\over y}+{y\over z}+{z\over x}$ is an integer. Find all possible values of $x+y+z$. Please provide a proof with all solutions.

Is $\mathbb{Z}$ the only totally-ordered PID that is “special”?

(All my rings are commutative and unital.) Definition. Call a totally-ordered ring $R$ special iff for all non-zero $b \in R,$ every coset of $bR$ has a unique element in the interval $[0,|b|).$ Motivation. This means that for any non-trivial principal ideal $bR$ of $R$, we have a natural bijective correspondence between $R/bR$ and $[0,|b|)$. […]

?+?+?=30 (1,3,5,7,9,11,13,15)

This question already has an answer here: What three odd integers have a sum of 30? [duplicate] 1 answer

Is there a domain “larger” than (i.e., a supserset of) the complex number domain?

I’ve been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this. In the domain of natural numbers, addition and multiplication always generate natural numbers, staying in the same domain. However subtraction of a large number from a smaller one needs to […]