It’s been a while since there has been any math on the blog, so I figured I’d share a recent (trivial) mathematical fact I came upon while passing the time. A less noble goal is that I hope some of you will find it interesting enough to think about it for a while. In other words, I’m too lazy to keep working on it but I hope some others will fall into my trap and let me know the answer.
Several nights ago, a friend of mine was procrastinating on the Internet — something I hear people do from time to time. He chanced upon a random (nerdy) web comic where the author declared that
I think basically everyone knows the Pythagorean triple
but I had never come across (or wondered privately) whether the pattern continued. A small group of us got together to determine an answer. Indeed, one can find consecutive integers whose squares add up to the sum of the next squares. For example,
In general, the problem is to find which solves the equation
Of course must depend on . Solving this equation is pretty simple using the following identities:
Using these identities correctly in our equation produces the polynomial
Plugging in produces
which forces the choice , i.e. . So, indeed, we can find a sequence of consecutive integers so that the squares of the first of them add up to the sum of the squares of the last .
Now, it’s also true that
It is an exercise left to the reader to figure out the general pattern for cubes, if it exists, and then generalize. Homework is due next week.