Just last week I received a request for a permission to use a picture from my Introduction to Graphs page in a forthcoming book. I gave the permission but noted that the picture is not mine and that I had lifted it from an old book years ago. None of the references listed on the page contained exactly that picture. Having no recollection where the picture came from, I began to search through my library. I have not found the picture yet, but did serendipitously come across several problems I have also forgotten.
Here's one such problem.
I made the formulation intentionally vague to enhance the effect of the coming clarification.
Quite obviously, in integers the only possible solutions are the combinations of 0s and 1s, like x = 1, y = z = 0, or y = 0, x = z = 1, etc. These solutions we may call trivial. Thus in integers the equation has only trivial solutions. However, this is not so is we allow x, y, z to be rational numbers. Furthermore, there is a really simple algorithm that reveals infinitely many solutions. And not only that. The algorithm has an obvious extension to similar equation in any number of variables, like
Let a, b, c be arbitrary integers. Introduce
On the other hand,
which is obviously the same.
For example, let a = 1, b = 2, c = 4. κ = 7 / 21 = 1/3. x = 1/3, y = 2/3, z = 4/3. It is easy to see that
The extension to any number of variables is indeed obvious. Further, we may reverse the process and ask for what κ the equation
holds for some a, b, c. Obviously, the answer is the formula for κ listed above:
It looks like the equation x + y + z = x³ + y³ + z³ or even when passing to higher powers all have infinitely many solutions in rational numbers. Further extension includes mixing the powers, like in
Once you get the idea, the original equation x + y + z = x² + y² + z² does not look as intimidating and mysterious as it did at the outset.
- B. A. Kordemsky, Mathematical Temptations, Publishing House ONIKS, 2000 (in Russian)