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What is this: a^2 + b^2 = c^2

The Pythagorean Theorem.

Right. What else is it?

An equation in three unknowns.

Do you know some solutions to this equation?

Sure. (3,4,5) and (5,12,13).

Right. Those are common pythagorean triplets. Do you know others?

Well, Google is my friend, let’s see. (typing) It looks like (7,24,25) and (9,40,41) all satisfy the equation.

Have you noticed that you’ve only supplied integer solutions?

Oh, right. I suppose that there are a whole load of non-integer solutions.

Have you heard of Diophantus?

Is that a Greek name?

Yes. Diophantus was interested in equations that had integral solutions. We call such equations: Diophantine equations.

So a^2 + b^2 = c^2 is a Diophantine equation?

Yes. And there are many others. For example: a^3 + b^3 = c^3

Oh, sure. And

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