How much structure can hide inside simple numerical patterns? Consider the following question: can three squares of rational numbers form an arithmetic progression? For example, 1², 5², 7² form an arithmetic progression, since 25−1=49−25=24. Are there infinitely many such examples? Is there a systematic way-- an algorithm - to produce them?
We then turn to a geometric problem: can we find a right-angled triangle with rational side lengths whose area is, say, 6? More generally, which numbers arise this way, and how can we construct such triangles?
At first glance, these questions seem unrelated—one about sequences, the other about geometry. However, we will see that they are intimately connected. Both lead naturally to Diophantine equations, where the goal is to find rational solutions to polynomial equations.
The aim of this talk is to illustrate how elementary questions can lead to rich mathematical ideas, and to invite the audience to think about whether systematic methods exist behind these patterns.
