The beauty of Diophantine equations

A Diophantine equation is usually an equation with two or more unknown variables, where their domain is the set of the integer numbers or a narrower set.

When I was in high school, I loved dealing with this kind of equations and there is a reason for that. In Diophantine equations, there’s no obvious way in how to even start. You can guess some numbers and if you’re lucky maybe those numbers satisfy the equality. However, that’s not enough. By guessing a solution, that doesn’t mean that it is unique.

The following problems are my 4 favorite problems from my notes of 100 problems collection in Diophantine equations:
CodeCogsEqn (4)

 

CodeCogsEqn (5)

 

CodeCogsEqn (3)

 

CodeCogsEqn (2) - Copy

The first three problems are from the book “An Introduction to Diophantine Equations” of T. Andreescu, D. Andrica, and I. Cucurezeanu.

And the last one is from the book “Problem-Solving Strategies” of A. Engel.